John 20:23 is not regarding forgiveness of sin from God. It is regarding forgiving others. A blood sacrifice is required for forgiveness of sin from God:
Hebrews 9:22
And almost all things are by the law purged with blood; and without shedding of blood is no remission.
God forgave the world of its sins through the shed blood of Jesus Christ:
2 Corinthians 5:18-21
And all things are of God, who hath reconciled us to himself by Jesus Christ, and hath given to us the ministry of reconciliation;
19 To wit, that God was in Christ, reconciling the world unto himself, not imputing their trespasses unto them; and hath committed unto us the word of reconciliation.
20 Now then we are ambassadors for Christ, as though God did beseech you by us: we pray you in Christ's stead, be ye reconciled to God.
21 For he hath made him to be sin for us, who knew no sin; that we might be made the righteousness of God in him.
Ephesians 1:7
In whom we have redemption through his blood, the forgiveness of sins, according to the riches of his grace;
Colossians 1:14
In whom we have redemption through his blood, even the forgiveness of sins:
Peter, James, John (the twelve) are trying to convince Israel of who Jesus Christ was (their Messiah). This is so they will receive their promised earthly kingdom of heaven and be forgiven of their post-cross sins at their "times of refreshing from the presence of the Lord".
Acts 3:12-21
And when Peter saw it, he answered unto the people, Ye men of Israel, why marvel ye at this? or why look ye so earnestly on us, as though by our own power or holiness we had made this man to walk?
13 The God of Abraham, and of Isaac, and of Jacob, the God of our fathers, hath glorified his Son Jesus; whom ye delivered up, and denied him in the presence of Pilate, when he was determined to let him go.
14 But ye denied the Holy One and the Just, and desired a murderer to be granted unto you;
15 And killed the Prince of life, whom God hath raised from the dead; whereof we are witnesses.
16 And his name through faith in his name hath made this man strong, whom ye see and know: yea, the faith which is by him hath given him this perfect soundness in the presence of you all.
17 And now, brethren, I wot that through ignorance ye did it, as did also your rulers.
18 But those things, which God before had shewed by the mouth of all his prophets, that Christ should suffer, he hath so fulfilled.
19 Repent ye therefore, and be converted, that your sins may be blotted out, when the times of refreshing shall come from the presence of the Lord.
20 And he shall send Jesus Christ, which before was preached unto you:
21 Whom the heaven must receive until the times of restitution of all things, which God hath spoken by the mouth of all his holy prophets since the world began.
Fortunately for us, through Israel's fall, salvation has come freely to all who will believe and without having to wait on Israel's "times of refreshing".
Romans 11:11
I say then, Have they stumbled that they should fall? God forbid: but rather through their fall salvation is come unto the Gentiles, for to provoke them to jealousy.
Romans 3:23-26
For all have sinned, and come short of the glory of God;
24 Being justified freely by his grace through the redemption that is in Christ Jesus:
25 Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God;
26 To declare, I say, at this time his righteousness: that he might be just, and the justifier of him which believeth in Jesus.