Your premis about who the letters was written to is partially off. When Paul and Peter wrote their letters, they were writing to Christian pastors who was to read the letters to their congregations made up of saved and unsaved people (just like today). As a result, the saved were buillt up in their faith, and the unsaved had an opportunity to see the need for salvation, and come to a relationship of faith in Christ. Looking at 1 John 1 in contex, you can see that it is addressing the lost. At the time John wrote this, it was around A.D 90, and a heretical group known as the Gnostics had infiltrated the ranks of the Christian orginizations.
One of the main teachings of the Gnostics was that man didn't have a sin nature, and even if he does, it doesn't matter. John addresses this in 1 John 1: 8 and 10. Looking at verse 9, we rarely look at what is said before and after:
Before:
8 If we say that we have no sin, we deceive ourselves, and the truth is not in us.
9 If we confess our sins, he is faithful and just to forgive us our sins, and to cleanse us from all unrighteousness.
After:
10 If we say that we have not sinned, we make him a liar, and his word is not in us.
So if we look at this in context, it is addressing those who believed that they had no sin. The Gnostics. The lost.