The Spirit and Jesus on Earth
The Gospels state the Holy Spirit was given at the baptism (Mark 1:9-11; Matthew 3:13-17; Luke 3:21-22). John records He remained on Jesus:
29 The next day he saw Jesus coming toward him, and said, “Behold, the Lamb of God, who takes away the sin of the world! 30 This is he of whom I said, ‘After me comes a man who ranks before me, because he was before me.’ 31 I myself did not know him, but for this purpose I came baptizing with water, that he might be revealed to Israel.” 32 And John bore witness: “I saw the Spirit descend from heaven like a dove, and it remained on him. 33 I myself did not know him, but he who sent me to baptize with water said to me, ‘He on whom you see the Spirit descend and remain, this is he who baptizes with the Holy Spirit.’ 33 And I have seen and have borne witness that this is the Son of God.” (John 1) [ESV]
Acts affirms Jesus had the Holy Spirit while He was alive:
how God anointed Jesus of Nazareth with the Holy Spirit and with power. He went about doing good and healing all who were oppressed by the devil, for God was with him. (Acts 10:38)
The Holy Spirit was instrumental in His Resurrection:
and was declared to be the Son of God in power according to the Spirit of holiness by his resurrection from the dead, Jesus Christ our Lord (Romans 1:4)
Therefore, any "loss" of the Spirit while on earth can only be a temporary condition, like the one after His birth and before His baptism. Likely this temporary separation is the significance of His cry before death:
And about the ninth hour Jesus cried out with a loud voice, saying, “Eli, Eli, lema sabachthani?” that is, “My God, my God, why have you forsaken me?” (Matthew 27:46)
Regardless, upon being raised from the dead He told Mary he had need to return to the Father which apparently was done immediately before He appeared to the other women:
Jesus said to her, “Do not cling to me, for I have not yet ascended to the Father... (John 20:17)
And behold, Jesus met them and said, “Greetings!” And they came up and took hold of his feet and worshiped him. (Matthew 28:9)
It is clear shortly thereafter He had the Spirit:
And when he had said this, he breathed on them and said to them, “Receive the Holy Spirit. (John 20:22)
So the Word coming into the world "disrupted" what had been "in the beginning" but at most, there are two periods during which He was not with the Holy Spirit: from birth until baptism and shortly before death until shortly after His Resurrection.
1 It is also important to note post-Resurrection descriptions of the Spirit are more intimate and personal than any description during His earthly mission. Where the Baptist saw the Spirit descend and remain, John reports the Spirit came from within Jesus, "He breathed on them..."
The disciples post-Resurrection experiences are described consistent with what Jesus had predicted:
But when the Helper comes, whom I will send to you from the Father, the Spirit of truth, who proceeds from the Father, he will bear witness about me. (John 15:27)
The lack of an intervening explanation of events between Jesus and His Father immediately after His Resurrection cannot negate the obvious: Jesus had the Spirit within Him and breathed Him upon the disciples soon after He was raised to life, exactly as He had predicted before His death. Therefore, Peter's post-Resurrection explanation of the events on Pentecost are not describing some new relationship with the Spirit, the Word, and God. Rather, it is affirming the condition which had been "in the beginning" but has been made available to all who receive Him and believe in His name (cf. John 1:12).
The Spirit with God
The Old Testament supports understanding the Holy Spirit departed from Jesus at His death:
and the dust returns to the earth as it was, and the spirit returns to God who gave it. (Ecclesiastes 12:7)
The spirit returns to God, not the Father. Any assumption that "Father" may be substituted for "God" is an anachronism based on an incomplete application of New Testament revelation:
In the beginning was the Word and the Word was with God... (John 1:1)
"God" cannot be replaced with Father without also considering the question of the presence of the Word with God. So if the Word had not yet been sent, the complete NT theological reading would be, the spirit returns to [the Word who was God and was with] God who gave it. At this time in history, the spirit is described as returning when the Word was still with God (and was God). There is no logical or Scriptural basis for separating the Word from God before He was sent.
Any attempt to understand "God" as "Father" must consider "in the beginning...the Word was with God, a condition which did not change until the Word came into the world. Therefore, if one replaces "God" with "Father" there must be a corresponding change with "the Word" and "Son."
...the spirit returns to God [the Father and Son] who gave it
This is no different from the present day state (cf. John 15:27). The present day sending of the Holy Spirit is the work of Father and Son just as any OT sending of the Spirit was a work of the Word with God. The difference is the New Testament reveals a more intimate connection between the Word and the Spirit then might be assumed possible had the Word not become flesh. That is to say, where the Old Testament describes the spirit returning to God who gave it; the New Testament describes the Word who was God breathing the Spirit into those who received the Word who had became flesh. So the emphasis is not on what returns to the giver, but what remains as a guarantee from the giver:
and who has also put his seal on us and given us his Spirit in our hearts as a guarantee.
(2 Corinthians 1:22) [cf. Ephesians 1:13, 4:30]
1 The significance of μόνον - alone ἀληθινὸν θεὸν in John 17:3 is a reflection both the Spirit and Son are "away" at the time Jesus is praying.