This is a contentious issue. Most notably, if for not other reason, than that it lies at some of the root of the difference between Protestants and Catholics.
Again Jesus said, "Peace be with you! As the Father has sent me, I am sending you." And with that he breathed on them and said, "Receive the Holy Spirit. If you forgive anyone’s sins, their sins are forgiven; if you do not forgive them, they are not forgiven."
Considering the verse above, there is actually a dispensation given to believers to forgive sins.
Now, before anyone goes off on any particular tangent, I believe there are many ways this comes about, but that they all point to Christ's sacrifice in the end.
Just as there are many different formulas for healing in the New Testament (in James, it is the elders anointing with oil and the prayer of faith, in the sending of people, it is the laying on of hands, in other places, it was faith, in others, the power of God present to heal). So, too, I see a variety of 'formulas' to bring about forgiveness. None are required, any will work.
When Jesus saw their faith, he said to the paralyzed man, “Son, your sins are forgiven.”
Now some teachers of the law were sitting there, thinking to themselves, “Why does this fellow talk like that? He’s blaspheming! Who can forgive sins but God alone?”
Immediately Jesus knew in his spirit that this was what they were thinking in their hearts, and he said to them, “Why are you thinking these things? Which is easier: to say to this paralyzed man, ‘Your sins are forgiven,’ or to say, ‘Get up, take your mat and walk’? But I want you to know that the Son of Man has authority on earth to forgive sins.” So he said to the man, “I tell you, get up, take your mat and go home.”
Going with the understanding that Jesus did everything as the Son of Man, and not the Son of God, it follows that he is speaking that a man, anointed by the Holy Spirit, can forgive sins. This would include believers, then.
Further, there is a rhetorical question. Which is easier? To forgive the man or heal him? The answer isn't given, but it seems to me that healing was easier.
What appears to be the case, however, is that, through the agency of the Holy Spirit, a believer can forgive sins, as well as heal the sick, cast out demons, and raise the dead. This of course points to Christ, as all forgiveness would.
With Christ Himself, of course, it would partly be of the virtue of Him being sinless, in the same way that He received the Holy Spirit in the first place (a well-pleasing Son). Once anointed, it was His calling to do so, as led by the Father (remembering that John had already prepared the way through repentance). However, this dispensation, in subjection to the will of the Father, was part of the calling, as described.
For us, we enter into that same perfection through His sacrifice, and participate in the same. Albeit, just as with healing, it is not apparent that it is necessary that forgiveness come through another person (say, a priest), but that it can. This forgiveness, then, too, would be said to come from Christ and His sacrifice, ultimately.
The notion that only God can forgive sins, however, seems directly what Jesus was saying wasn't the case in Mark 2:6-9.