Therefore let us also, having so great a cloud of witnesses surrounding us, put away every encumbrance and the sin which so easily entangles us and run with endurance the race which is set before us.
My understanding of 12:1 is that Paul, whom I feel is the author, is talking about the particular sin which entangled all these Hebrew believers to whom he wrote from Miletus towards the end of his race and life. I don't feel he's talking about sin in general, or various believers' "besetting" sins.
I'll try to quote portions that elaborate on my thought that the epistle to the Hebrews, as a whole, is directed against the sin of Judaism. That is, the sin of Judaizing (Ac 21:20-26; Gal 2:11-16). It's a massive topic which can engage most books of the New Testament, but I'll try to be concise. The author alternates between showing the superiority of Christ to the old things (maybe as he does in all his letters), and then warning the believers not to miss the prize and suffer discipline in the next age (cf 1 Cor 3:10-17). Needless to say there's nothing about eternal damnation here since he's writing to Christians, believers
God, having spoken of old in many portions and in many ways to the fathers in the prophets, has at the last of these days spoken to us in the Son, whom He appointed Heir of all things, through whom also He made the universe; who, being the effulgence of His glory and the impress of His substance and upholding and bearing all things by the word of His power, having made purification of sins, sat down on the right hand of the Majesty on high. Having become as much better than the angels...
For He has been counted worthy of more glory than Moses, by as much as He who built the house has more honor than the house.
Having therefore a great High Priest who has passed through the heavens, Jesus, the Son of God, let us hold fast the confession. For we do not have a High Priest who cannot be touched with the feeling of our weaknesses, but One who has been tempted in all respects like us, yet without sin...High Priest according to the order of Melchisedec...[T]hrough Abraham, Levi also, he who receives tithes, has been made to pay tithes, for he was still in the loins of his father when Melchisedec met him. If indeed then perfection were through the Levitical priesthood (for under it the people have received the law), what need was there still that a different Priest should arise according to the order of Melchisedec and that He should not be said to be according to the order of Aaron?
[T]he setting aside of the preceding commandment because of its weakness and unprofitableness (for the law perfected nothing)...Jesus has also become the surety of a better covenant...Who does not have daily need, as the high priests do, to offer up sacrifices first for his own sins and then for those of the people; for this He did once for all when He offered up Himself. For the law establishes men as high priests who have weakness, but the word of the oath, which was after the law, establishes the Son, perfected forever...He has obtained a more excellent ministry inasmuch as He is also the Mediator of a better covenant, which has been enacted upon better promises. For if that first covenant had been faultless, no place would have been sought for a second. For finding fault with them He says, "Behold, the days are coming, says the Lord, and I will consummate a new covenant upon the house of Israel and upon the house of Judah"...In saying, A new covenant, He has made the first old. But that which is becoming old and growing decrepit is near to disappearing.
Christ, having come as a High Priest of the good things that have come into being, through the greater and more perfect tabernacle not made by hands, that is, not of this creation, and not through the blood of goats and calves but through His own blood, entered once for all into the Holy of Holies, obtaining an eternal redemption. For if the blood of goats and bulls and the ashes of a heifer sprinkling those who are defiled sanctify to the purity of the flesh, how much more will the blood of Christ, who through the eternal Spirit offered Himself without blemish to God, purify our conscience from dead works to serve the living God?
Wow, that's a lot. I better try to jump to the sin question. The personal sin question. Although Judaizing is also a "personal" sin. They opposed Paul and damaged the other saints. "I wish that those upsetting you would even cut themselves off......Beware of the dogs, beware of the evil workers, beware of the concision." Gal 5:12; Philip 3:2. One can be conscious of the wrong, the sin, of Judaizing, as an objective problem, and enjoy being forgiven of or delivered from it. Like Paul himself was, you could say broadly. "You have heard of my manner of life formerly in Judaism, that I persecuted the church of God excessively and ravaged it......For I am the least of the apostles, who am not fit to be called an apostle, because I persecuted the church of God." Gal 1:13; 1 Cor 15:9; cf Rv 3:9. Paul was forgiven of that, justified by faith.
So I'm saying that the consciousness of guilt of sin in one's conscience is what's spoken of in Hebrews 10:2. Which Jesus' blood, praise the Lord, effectively, eternally washed and washes away.
The law, having a shadow of the good things to come, not the image itself of the things, can never by the same sacrifices year by year, which they offer continually, perfect those who draw near. Otherwise would they not have ceased to be offered, because those worshipping, having once been purified, would have no longer had the consciousness of sins? But in those sacrifices there is a bringing to mind of sins year by year; for it is impossible for the blood of bulls and goats to take away sins. Therefore, coming into the world, He says, "Sacrifice and offering You did not desire, but a body You have prepared for Me. In burnt offerings and sacrifices for sin You did not delight. Then I said, Behold, I have come (in the roll of the book it is written concerning Me) to do Your will, O God"...He takes away the first that He may establish the second, by which will we have been sanctified through the offering of the body of Jesus Christ once for all. And every priest stands daily, ministering and offering often the same sacrifices, which can never remove sins; but this One, having offered one sacrifice for sins, sat down forever on the right hand of God, henceforth waiting until His enemies are made the footstool for His feet. For by one offering He has perfected forever those who are being sanctified. And the Holy Spirit also testifies to us, for after having said, "This is the covenant which I will covenant with them after those days, says the Lord: I will impart My laws upon their hearts, and upon their mind I will inscribe them," He then says, "And their sins and their lawlessnesses I shall by no means remember anymore." Now where forgiveness of these is, there is no longer an offering for sin.
The blood of Jesus His Son cleanses us from every sin. If we say that we do not have sin, we are deceiving ourselves, and the truth is not in us. If we confess our sins, He is faithful and righteous to forgive us our sins and cleanse us from all unrighteousness...[I]f anyone sins, we have an Advocate with the Father, Jesus Christ the Righteous; and He Himself is the propitiation for our sins, and not for ours only but also for those of the whole world.
[The Son] having made purification of sins, sat down on the right hand of the Majesty on high.
Who does not have daily need, as the high priests do, to offer up sacrifices first for his own sins and then for those of the people; for this He did once for all when He offered up Himself.
I will be propitious to their unrighteousnesses, and their sins I shall by no means remember anymore. Heb 8:12.
Not through the blood of goats and calves but through His own blood...purify our conscience from dead works to serve the living God