Ancient People Believed Many Things
It's probable that the prevailing worldview of Israel during the time of Saul was that the dead are, in some manner or another, conscious and capable of being contacted(as shown in @HoldToTheRod's answer). Is it a problem that they believed that? Does that mean it's true or that we should also believe what they believed? Not necessarily. Throughout the ages, the Israelites believed many things that, in retrospect, were plainly incorrect (they were an ancient civilization, after all). For example, they believed in a highly inaccurate cosmology vastly distinct from the one we have today. They believed that the earth was flat and circular (not spherical), having foundations underneath in the vast abyss of the deep (Isaiah 40:22 [cf. Isaiah 22:18, where the actual Hebrew word for a spherical object is used], Job 9:6, Job 26:10, Psalm 75:3, Psalm 102:25, Proverbs 8:27). They believed in a solid firmament (called heaven) that kept the waters above the sky separated from the waters below the sky so that land could exist, and that there were windows in this firmament that can be opened to flood the land, and that this firmament had foundations supporting it (Genesis 1:7-8, Genesis 7:11, Genesis 8:2, 2 Samuel 22:8, Isaiah 24:18, Job 26:11, Job 37:18, Psalm 148:4-6, Ezekiel 1:22-26). They also believed that the kidneys, not the brain, were the center of thought and feeling (Psalm 7:9, Psalm 16:7, Psalm 26:2, Psalm 73:21, Proverbs 23:16, Job 16:12-13, Jeremiah 11:20, Jeremiah 17:10, Lamentations 3:13, Revelation 2:23).1
Why Not Correct Them?
So why didn't God ever correct them on these matters? Why didn't He tell them that the Earth is actually a sphere? That there are no waters above the sky? That there are no windows in heaven? That the brain, not the kidneys, is actually the center of thought and feeling? Because being ignorant of the truth is part of human nature, and when not pertinent/significant to our relationship with Him, God is not required to help us out of our ignorance. That's not to say, of course, that God never helped us out of our ignorance when it wasn't necessary. God revealing the truth to us about the consciousness of the dead is just one example of God helping us out of our ignorance when not pertinent to our relationship with Him. The point is, what Israel's views were on the consciousness of the dead was not very pertinent to their relationship with God; it's not very pertinent to our relationship with God (that Israel had faith in Yahweh and kept His commandments was most essential; that we have faith in Yeshua and keep His commandments is most essential).
So yes, Saul and his contemporaries most likely believed that the dead were conscious in some manner or another, and that they could be contacted. They also believed that Yahweh sent evil spirits to torment people (1 Samuel 16:14-16). Does this indicate that their belief that the dead were conscious and capable of being contacted (or that God sent evil spirits to torment people) was true? Not necessarily. The fact that Saul and his contemporaries believed that the dead were conscious does not in any way necessitate that the dead are actually conscious. David, on the other hand, possibly believed that the dead were unconscious, or at the bare minimum knew that the dead were fully incapable of having any sort of association with the living whatsoever, and vice versa.
God Progressively Reveals Things
What do I mean when I say that God progressively reveals things? Do I mean that newer revelations can contradict and replace/nullify older revelations? No. God's revelations will always be thoroughly consistent with each other; successive revelations necessarily produce a fully coherent framework (we will see that here apropos the state of the dead). Yes, Saul and his contemporaries most likely believed that the dead were conscious and capable of being contacted (well Saul undoubtedly), but why did they? Well, because God never revealed to them, or their predecessors, the truth about the state of the dead! I cannot find a single statement made by Yahweh (or any of His inspired writers/prophets), made before or during the time of Saul, that speaks about the state of the dead (about their consciousness or lack thereof). How do you expect the Israelites (during the time of Saul) to know that the dead are unconscious and thus incapable of being contacted... if God hadn't revealed to them that it is so?
It's pretty straightforward; God had not yet revealed to them the truth about the state of the dead. Eventually, God did reveal the complete truth about the state of the dead (in Ecclesiastes), i.e. that the dead are thoroughly unaware and unconscious; but this was long after Saul (and David) had passed away. The majority of scholars date the writing of the book of Ecclesiastes to somewhere around 250 B.C., around 750 years after Saul would have lived! Neither Saul nor David could have read Ecclesiastes, hence how could they have known for a fact that the dead were unconscious? Although I must say, David (and others inspired by God to write Psalms), had a much clearer picture of the state of the dead; I cannot say for sure if he knew that they were fully unconscious/unaware. Perhaps, perhaps not. Either way.
Here's The #1 Point
Here is, by far, the most significant point: While God might have allowed the Israelites to believe that the dead were conscious (even allowing some such as Saul and his contemporaries to believe that they were capable of being contacted), you know what's one thing God never once allowed them to believe? That the righteous go to heaven and the unrighteous go to hell; more accurately, that the righteous go to one compartment of Sheol to be in bliss and the unrighteous go to another compartment of Sheol to be in torment. In fact, quite the opposite! Whatever the Israelites did believe about the dead, it utterly opposes any notions of heaven and hell (or individual compartments of Sheol, one dedicated for the righteous who live there in bliss, and another dedicated for the unrighteous who live there in torment).
I want you, for one moment, to completely forget everything you've ever thought, believed, or assumed about the state of the dead. If you were to then read each of the following passages, would you ever in a billion years come out with the notion that the dead are either being tormented in hell (compartment of Sheol for the unrighteous) or experiencing bliss in heaven (compartment of Sheol for the righteous, i.e. paradise/Abraham's bosom)? We will see that such notions are thoroughly foreign to the Bible.
Genesis 3:9 By the sweat of your brow you will eat your bread, until you return to the ground—because out of it were you taken. For dust you are, and to dust you shall return.” (What does God say the fate of humankind is? To return to the dust. Does dust remind you of noise or activity [of which there would be in heaven or hell], or does it remind you of silence and lifelessness? Also, notice how God does not say, "until your body returns to the ground, because out of it was your body taken; for dust your body is, and to dust it will return." WE go to the ground, not our bodies; we do not go to be with God, but our spirits [i.e. the spirits that belong to us, not the spirits that are us] go to be with God.)
Psalm 6:4-5 Turn, O Yahweh, deliver my life; save me for the sake of your steadfast love. 5 For in death there is no remembrance of you; in Sheol who will give you praise? (David really does not want to be dead. He wants God to deliver his life, for he fully believes that, while dead, he is incapable of praising or remembering God.)
Psalm 9:5 You have rebuked the nations, You have eliminated the wicked; You have wiped out their name forever and ever. (Has God sent the wicked to eternal torment in fire for eternity or has He eliminated them? You can't have both. David clearly goes with the latter.)
Psalm 16:10 For you will not abandon my soul to Sheol, or let your holy one see corruption. (David seems as though he really doesn't want to be in Sheol. But why not? Would he not be in heavenly bliss in Abraham's bosom/paradise?)
Psalm 21:9 You will place them in a fiery furnace at the time of Your appearing. In His wrath Yahweh will engulf them, and the fire will consume them. (What image comes to mind when you hear about fire consuming something? If you were to throw, say, a notebook into a bonfire, do you expect that the notebook would stay intact [even if just partially] for eternity? Or do you expect the notebook to be demolished; reduced to ashes? The fire of Yahweh will demolish the wicked because that is what fire does; it demolishes!)
Psalm 22:15 My strength is dried up like a potsherd, and my tongue sticks to the roof of my mouth. You lay me in the dust of death. (Here's dust being metaphorical of death again. Notice how David says, "you lay ME in the dust of death", and not, "you lay my body in the dust of death". Once again, a human person is said to be in the dust; like dust, silent without any activity.)
Psalm 30:9 “What profit is there in my death, if I go down to the pit? Will the dust praise you? Will it tell of your faithfulness? (The theme of "dust" being lifeless and silent is being hammered down once more. I'd say that if I went to paradise to be with God upon death, there would be profit in my death, no?)
Psalm 31:17 O Yahweh, let me not be ashamed, for I have called on You. Let the wicked be put to shame; let them lie silent in Sheol. (Notice how David does not say, "let them lie in torment in Sheol", rather he says, "let them lie silent in Sheol". Torment was not anywhere near David's mind when he thought about the wicked dead; quite the opposite! I hardly imagine that you can be silent while being tormented by fire day and night...)
Psalm 37:20 But the wicked will perish; the enemies of Yahweh are like the glory of the pastures; they vanish—like smoke they vanish away. (The wicked will perish and vanish like smoke? It is beyond unimaginable that one could ever read eternal torment into this passage.)
Psalm 37:38 But wrongdoers will altogether be destroyed; The future of the wicked will be eliminated. (As you can see, transgressors will be tormented; they will have an eternal future in hellfire. Wait, my bad, that's not what it says. It says that wrongdoers will be destroyed, and subsequently so too any future they might have. Being tormented in hellfire for eternity certainly seems like a future to me.)
Psalm 49:15 But God will ransom my soul from the power of Sheol, for He will receive me. Selah (If when the righteous die and go to Sheol, they go to be with God, why is it that God needs to ransom a person's soul from Sheol in order to receive them? It's almost as if God is not in Sheol.)
Psalm 88:10-12 Do you perform wonders for the dead? Do the departed rise up to praise you? Selah 11 Is your steadfast love declared in the grave, or your faithfulness in Abaddon? 12 Are your wonders known in the darkness, or your righteousness in the land of forgetfulness? (Why doesn't God perform wonders for the dead? When Lazarus died [Luke 16:19-31], God undoubtedly performed wonders for him; He sent His angels to bring him up to the bosom of Abraham! But if the righteous dead are in paradise with God, why doesn't He perform wonders for them. Is it perhaps that they are incapable of perceiving them? This fits perfectly with what is said afterward; not only are God's wonders not known in the darkness of death, but neither is His love, nor His faithfulness, nor His righteousness; nothing is known in the land of forgetfulness.)
Psalm 90:3 You return man to dust and say, “Return, O children of man!” (Who returns to dust? Man, not man's body, returns to dust.)
Psalm 94:16-17 Who will rise up for me against the wicked? Who will stand for me against the workers of iniquity? Unless Yahweh had been my helper, I would soon have dwelt in the abode of silence. (The Psalmist is saying that if Yahweh had not protected him from the wicked, he would have soon died, or in his words, "dwelt in the abode of silence." The dead dwell in the abode of silence. That does not remind me of either heaven or hell, or should I say, one compartment for the bliss of the righteous, and another for the torment of the wicked. The Psalmist presents death as a place with no noise or activity.)
Psalm 94:24 He has brought back their injustice upon them, And He will destroy them in their evil; Yahweh our God will destroy them. (God has brought the injustice of the wicked upon them, and will torment them because of their evil; He will torment them for eternity. Oh wait, my bad again, that's not what it says.)
Psalm 103:14 For He knows our frame; He is mindful that we are dust. (Once more, WE are dust; not our bodies are dust.)
Psalm 104:29 When you hide your face, they are dismayed; when you take away their spirit, they die and return to their dust. (Notice how a distinction is made between us and our spirit; when our spirit [i.e. spirit that belongs to us, not is us] is taken away, we die and return to dust. Not, when we are taken away, our bodies die and return to dust. We are not our spirits, and our bodies are not what return to dust; we are what return to dust.)
Psalm 115:17 The dead do not praise Yahweh, nor do any who descend into silence. (This Psalm is almost a combination of Psalm 6:5 and Psalm 94:17. Death is once more said to be silent, with no praises of Yahweh coming out from it.)
Psalm 146:4 When his spirit departs, he returns to the ground; on that very day his plans perish. (Notice how the Psalmist does not say that we depart, and that our body returns to the ground. It says our spirit [i.e. the spirit that belongs to us, not is us] departs, and that we [us, human persons] return to the ground. Some would have it say, "when he departs, and his body returns to the ground"; though it says the exact opposite! The spirit that belongs to man [not the spirit that is man] is what departs, and the man himself returns to the ground whence he came from. It is that clear and unequivocal.)
Isaiah 38:17-18 Surely for my own welfare I had such great anguish; but Your love has delivered me from the pit of oblivion/nothingness, for You have cast all my sins behind Your back. For Sheol cannot thank You; Death cannot praise You. Those who descend to the Pit cannot hope for Your faithfulness. (The pit of oblivion and nothingness? Oblivion is defined as "the state of being unaware or unconscious of what is happening." Nothingness is defined as "the absence or cessation of life or existence." How much more explicit about death does Isaiah need to be? Does the next verse work? Isaiah says that the dead cannot thank, cannot praise, and cannot hope... Do I have faith that Yeshua will deliver me from the pit of oblivion!)
Ecclesiastes 3:19-21 For the fates of both men and beasts are the same: As one dies, so dies the other—they all have the same breath. Man has no advantage over the animals, since everything is futile. All go to one place. All are from the dust, and to dust all return. Who knows whether the spirit of man goes upward and the spirit of the beast goes down into the earth? (Do all bodies return to the dust or do all creatures return to the dust? Once more, we human beings are dust; not our bodies. We return to the dust; not our bodies. The spirits that belong to us are what rise upwards to God for safekeeping [cf. Ecclesiastes 12:7, Luke 23:46, Acts 7:59], but our spirits are not synonymous with our very being; they are our possession. It's the spirit OF man, not the spirit THAT IS man.)
Job 10:9 Please remember that You molded me like clay. Would You now return me to dust? (Does Job say, "would you now return my body to dust"? No, he says, "would you now return ME to the dust." Once more, human beings, not bodies, return to the dust. We are dust; not our bodies. There is not a single place in all of scripture where the dust metaphor is used in conjunction with our bodies, but always with our very being, our entire existence as human persons.)
Job 14:10-15 But a man dies and lies prostrate. A person passes away, and where is he? 11 As water evaporates from the sea, And a river becomes parched and dried up, 12 So a man lies down and does not rise. Until the heavens no longer exist, He will not awake nor be roused from his sleep. 13 “Oh that You would hide me in Sheol, That You would conceal me until Your wrath returns to You, That You would set a limit for me and remember me! 14 If a man dies, will he live again? All the days of my struggle I will wait Until my relief comes. 15 You will call, and I will answer You; You will long for the work of Your hands. (What conception do the words "water evaporates from the sea" and "river becomes parched and dried up" conjure in your mind? Noise and activity? Bliss or torment? Or do you envision decimation and stillness? Job is analogizing such a scenario with the circumstances of those who die. The dead are "dried up and parched" and "evaporated", i.e. still and inactive. That kind of reminds me of a person in deep sleep. Those who are asleep aren't up about doing anything; they aren't making conversation, solving differential equations, or cooking a meal. In fact, they're completely unaware of their surroundings. Perhaps that is why Job compares death to sleep. Because the dead are in an inactive, motionless, unaware state; just as those asleep are. Many don't realize just how absurd and meaningless the analogy of death and sleep becomes when you interpret "asleep" to mean experiencing either bliss in a heavenly paradise with God or day-and-night torment in an unquenchable fire; could you be anywhere near asleep in either scenario? I've heard some argue for Job disbelieving in the possibility of a resurrection. It is beyond inconceivable that Job would not have believed in the possibility of a resurrection judging by his words in verse 15; "you will call and I will answer." Does that remind you of anything? It reminds me of John 5:25 and 28-29, where those who hear the voice of Yeshua [i.e. Yeshua calls them] will come out of the tombs and live [i.e. they answer His voice by coming to life]. Job had faith that if God were to call him from his state of death, Job would answer and come to life. The only thing Job is wondering about is whether or not God WILL do that, not whether or not God CAN do that.)
Job 17:13-16 If I hope for Sheol as my house, if I make my bed in darkness, 14 if I say to the pit, ‘You are my father,’ and to the worm, ‘My mother,’ or ‘My sister,’ 15 where then is my hope? Who will see my hope? 16 Will it go down to the bars of Sheol? Shall we descend together into the dust?” (Sheol is depicted as darkness, with no hope in it. Job does not expect to go into the dust along with his hope. But notice that Job does not say that he has no hope at all, but simply that the hope that he does have does not go with him into the pit. Job is not denying hope; he's denying that hope goes along with him into Sheol. While the hope that you and I have does not go with us into Sheol, it does not need to, as the hope that we have in Yeshua transcends both life and death [cf. Romans 14:8, Philippians 1:20, 1 Thessalonians 5:10].)
Job 34:14-15 If He were to determine to do so, If He were to gather His spirit and His breath to Himself, 15 Humanity would perish together, And mankind would return to dust. (This is perhaps one of the most unambiguous and incontrovertibly straightforward statements needed to obliterate any notions of spirits being man or bodies going to the dust. If God were to remove the vital life-sustaining spirit and breath of mankind, mankind would perish altogether and return to the dust. It does not say that we would go to God; somehow people take Ecclesiastes 12:7 to mean that we humans go to God upon death, even though nowhere in all of scripture are we [us, human persons, living beings] equated to our spirit/breath [that which gives vitality and sustains a human being]. It also does not say that our bodies/only the physical part of us would return to the dust; it unequivocally says mankind would return to the dust.)
Daniel 12:2 And many of those who sleep in the dust of the earth shall awake, some to everlasting life, and some to shame and everlasting contempt. (Many of those who are experiencing either bliss in paradise with God or being tormented in unquenchable fire will awake, some to... wait, I feel like I made an error somewhere. Oh, it says that many of those who sleep in the dust of the earth shall awake. Do the words "dust" and "sleep" conjure up notions of heavenly bliss or day-and-night torment? Or do they conjure up notions of quietude and dormancy; silence and inactivity? How one can interpret "sleep in the dust" as "bliss in paradise with God" or "unending torment in fire" is entirely beyond me. Daniel 12:2 makes two clear statements; (1) that the dead are asleep and (2) that they are in the dust of the earth. Both of these are metaphors for the state of the dead. The dead are "asleep" because, like those asleep, they are unaware of themselves and their surroundings; the dead are "in the dust of the earth" because, like the dust of the earth, they are silent and lifeless. It's as straightforward as that; there's no equivocation or ambiguity.)
As you can see, while God may have allowed the Israelites during the time of Saul to believe in the consciousness of the dead, there is one thing He never allowed them to believe; that there are separate compartments in Sheol, one for the righteous where they will experience heavenly bliss with God and another for the unrighteous who will be tormented in fire with no relief. Such notions would have been entirely foreign to the mind of any Israelite, whether Saul or David.
See How It All Comes Together?
I had also said something else before; "successive revelations necessarily produce a fully coherent framework (we will see that here apropos the state of the dead)." Have we seen that? 100%! Remember, Saul, who lived in the 11th century BC, was very limited in his knowledge about the state of the dead, that is, the knowledge he had was completely incorrect (he believed not only that Samuel was conscious but that he was capable of being contacted). David, living in the 10th century BC (at least for 30 years, until he died in 970 BC), would have had much more knowledge, and he certainly did judging by some of the psalms he wrote. But did he fully know the state of the dead? I cannot say for sure. Isaiah was written during the 8th century BC (300 years after Saul died). And Job was written around the 6th century BC (500 years after Saul died).
All of the passages we read (excluding the one in Daniel) were written before Ecclesiastes, which was most likely written in the second half of the 3rd century BC (750 years after Saul died). Judging solely by what we have read so far, one can very easily guess that the dead are unconscious. But God's purpose is not to leave us guessing. His purpose is to make sure we have the full, complete truth about the state of the dead. It's as though everything that has been said so far has been building up to the finale; the pinnacle of it all! God is about to reveal the whole truth, one that will, along with everything else we've learnt so far, construct a fully coherent framework. God is going to reveal something that is 100% consistent with all preceding revelations. Let's read it.
Ecclesiastes 9:5-6 For the living know that they will die, but the dead know nothing, and they have no more reward, for the memory of them is forgotten. 6 Their love and their hate and their envy have already perished, and forever they have no more share in all that is done under the sun.
Ecclesiastes 9:10 Whatever your hand finds to do, do it with your might, for there is no work or thought or knowledge or wisdom in Sheol, to which you are going.
That's why there is no remembrance of God in death! Why His wonders, love, faithfulness, and righteousness are not known; the dead do not know anything! That's why death is called the land of forgetfulness; no one remembers anything, because they are without knowledge or thinking ability. That's why the dead are said to be dust; they are silent and lifeless. It's why the state of the dead is analogized to sleep; they are dormant and inactive. It's also why the dead cannot praise, cannot thank, and cannot hope; they are fully unconscious, incapable of performing such feats which require mental faculties. It's why those who die descend into silent darkness, and "dwell in the abode of silence"; there is no activity or noise or work for those who are dead, as they are fully unaware of anything. It's why David hated the thought of going to Sheol; I'd certainly hate being totally lifeless and unconscious/unaware. This is also why death is called the "pit of nothingness/oblivion"; because the dead are nonexistent, fully unaware of anything that is happening. Everything fits together perfectly, like pieces of a puzzle. Ecclesiastes was the last puzzle piece needed to construct a fully articulate conception about the state of the dead. There is nothing more to be done.
In Conclusion:
Yes, Saul and David had different afterlife theologies. One (David) was closer to the truth than the other (Saul), though neither had the complete truth. Were there different competing afterlife theologies on sale in the "spiritual marketplace" of that time? There are competing theologies about any subject on sale at any time, perhaps more so in our day than ever before (that's consistent with the fact that there are nearly 8 billion people living on this planet)! Although, in the case of Israel, it's not so much that there were competing theologies as it is that there was simply very limited knowledge at that time about the state of the dead. In all the books preceding 1 Samuel, there is not a single statement about the state of the dead (i.e. about their consciousness or lack thereof). There are statements about where the dead go (e.g. Genesis 37:35, Genesis 42:38, Genesis 44:29, Genesis 44:31, Numbers 16:30, Numbers 16:33). But none about their current state of consciousness. That explains why Saul believed that Samuel (who had died) was capable of being contacted, and why his contemporaries didn't view Saul's belief as anything out of the ordinary.
No matter what, we are not obliged to believe the same things Saul and his contemporaries believed, simply because they believed them; otherwise, we'd have to believe that God sends evil spirits to torment people (1 Samuel 16:14-16), and that the earth is flat and circular with foundations in the abyss of the deep, and that there is a solid firmament in the sky, keeping the waters above from drowning the land, and that the kidneys, not the brain, are the center of thought and emotion. Of course, that Israel believed all these things is not in any way a problem, as it wasn't greatly significant to their relationship with God. Being ignorant of the truth is ingrained into our nature, our imperfect, sinful nature; and God knows this, and He does not hold it against us, which is why we should be all the more thankful when God reveals the truth to us!
Hope this helps!
Notes:
1 I realize that some may be confused about my particular view of the Bible. I don't believe in Biblical inerrancy; if by inerrancy one means that the Bible is incapable of ever making a single historical or scientific mistake. I do, however, believe that the primary messages conveyed by the Bible are always fully true, that is, that the essence of what the Bible teaches is completely accurate. For example, the Genesis account states that God created everything in the universe in 6 days. Do I believe that God created everything in the universe in 6 days? No. Do I believe in the message that the Genesis account is trying to convey, namely, that God is the Ultimate Creator of all things, the One who, through the power of His Word, brought forth an ordered reality for His creation to live in and enjoy? I do 100% and I don't doubt it! Likewise, when Jesus says that He is the One who searches the kidneys and hearts, do I believe that Jesus actually searches a person's kidneys to find out their thoughts? No. Do I believe the essence of Jesus' teaching here, namely, that He is the One who searches and comprehends the emotions and thoughts of man? Again, I do 100% and I don't doubt it!