From "Present, Eternal and Governmental forgiveness of sins." by J.N. Darby:
There is only another passage, which it may be well to refer to, John 20: 23. The Lord, after He was risen, comes amongst His disciples and communicates to them the peace He had just made, and sends them out to preach that peace to others while He has gone away into heaven. In thus sending them out as His Father had sent Him, He conferred on them apostolic authority, so that they should administer this remission and forgiveness of sins to all those who believed, who became Christians. Thus when the Jews, convinced of their sin in rejecting Christ, said, thinking all was over through their rejecting Him, "Men and brethren, what shall we do?" Peter replies, "Repent, and be baptised every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost." Thus becoming Christians through his ministry they received the perfect remission which Christ obtained for them.
So in Acts 10: 43; only there, as Peter himself had great difficulty in receiving any who believed from among the heathen, God gave them a testimony, as Peter says afterwards, before they were received, so that men could not refuse to receive them. So Paul gives the same testimony; Acts 13: 38, 39.
And to this day, if a heathen believes in Christ and becomes a Christian by baptism, he then receives full remission of sins. Only the apostles could do it, with not only personal authority, but discernment as to the reality of the faith of those who came; Acts 8: 28, 29. The general truth remains sure, "By him all that believe are justified from all things."
The same governmental forgiveness remains true, too, with the same difference. Peter does not pray for Ananias and Sapphira: it was a sin unto death, and they fell dead. So Paul (1 Cor. 5: 3-5) judged to "deliver such an one to Satan for the destruction of the flesh." There was apostolic discernment, authority and power. Then all were called on too to act on their responsibility as the assembly of God (vv. 12, 13), and Paul associates them with him in this act of power.
365 In this sense the apostles had no successors. There were local authorities, elders, deacons, etc., but apostles were apostles, and did this with Christ's authority everywhere, only Peter specially among the Jews, and Paul among the Gentiles; Gal. 2: 7, 8. As such, even the greatest authorities among the Roman Catholics admit they had no successors.
But there is a succession owned of God, whose authority flows from Christ where His presence is realised in lowly grace. In Matthew 18: 18, if a person were wronged, he was to speak to the wrong-doer, and win him if he could; if he could not, to take two or three more; if this did not succeed, he was to tell it not to the clergy, not to any priest, but to the assembly. If the wrong-doer would not listen to the assembly, the person was free to treat him as outside it as a heathen man and a publican. And the reason is given: that wherever two or three were gathered together in Christ's name, really met and looking to Him, so as to act really and humbly in His name, He being there according to this promise, the act would have in ordinary church discipline (as "putting out from among yourselves") Christ's authority, and He would own and sanction it.
It is not individual apostolic power (Peter and Paul both announce these would not be after their decease, Acts 20: 29-33) acting in Christ's name as Peter could, saying, "Jesus Christ maketh thee whole," or Paul delivering to Satan, but an act within the limits of duty, presented by the word, and which Christ sanctions by His presence and authority, acting in the midst of two or three. This supposes they are in unity, really gathered to Christ's name, and truly looking to Him by the Spirit, as the only One who can exercise this authority, and taking His word for their guide. It is this that in the word of God takes the place (I do not say of apostolic power, for it is not individual, but) of apostolic authority, because it is Christ who really acts.