I've read that to confess is to "name it as He names it". In other words, it is an acknowledgement of the truth of our act before God, who sees all and knows all. When we confess our sins, as I John 1:9 says, "God is faithful to forgive us our sins and to cleanse us from all unrighteousness". When our sin is committed against another person, part of that confessing is going to the person we harmed and admitting our act, asking them for their forgiveness. In some cases, restitution is part of it to restore to the person we sinned against what we took from them. The act of confessing our sins "one to another" is when our action has brought harm to the faith community and our confessing and asking forgiveness is a way God uses to restore fellowship and establish accountability one to another.
I believe this is how the early believers operated in their local assemblies according to what the epistles say. The practice of "auricular" (to the ear) confession to a priest was not around until the seventh century. From http://www.justforcatholics.org/a23.htm:
There was no auricular confession to a priest in the early church either. Augustine gives us a snapshot of the church in the 4th and 5th century. In his Sermon to Catechumens on the Creed, Augustine writes:
How did Christians deal with sin at that time? They dealt severely with those who committed grievous sins, casting them out of the church. A period of "penance" was required before the repentant sinner was re-admitted. But what about the daily sins that all Christians commit? Did they confess them to a priest? No, they confessed directly to God in prayer, asking the Father for forgiveness. Prayer was considered sufficient for daily cleaning.
The Catechism of the Catholic Church admits that private confession first came on the scene in the seventh century:
“Over the centuries the concrete form in which the Church has exercised this power received from the Lord has varied considerably. During the first centuries the reconciliation of Christians who had committed particularly grave sins after their Baptism (for example, idolatry, murder, or adultery) was tied to a very rigorous discipline, according to which penitents had to do public penance for their sins, often for years, before receiving reconciliation. To this ‘order of penitents’ (which concerned only certain grave sins), one was only rarely admitted and in certain regions only once in a lifetime. During the seventh century Irish missionaries, inspired by the Eastern monastic tradition, took to continental Europe the ‘private’ practice of penance, which does not require public and prolonged completion of penitential works before reconciliation with the Church. From that time on, the sacrament has been performed in secret between penitent and priest. This new practice envisioned the possibility of repetition and so opened the way to a regular frequenting of this sacrament. It allowed the forgiveness of grave sins and venial sins to be integrated into one sacramental celebration. In its main lines this is the form of penance that the Church has practiced down to our day” (Catechism of the Catholic Church, paragraph 1447).
We Catholics find ourselves with a new "custom", that of confessing "face to face" with a priest, I've always held that if I can look another man in the eyes and tell him my sins I'm not repenting I'm bragging.
Your source material is spot on about how serious sin was dealt with in the early days. As with all things however, once the size of the Christian assemblies grew larger than a modern day support group the process had to be streamlined. It's just not practical in a congregation of 2 or 3 thousand to have everyone confessing their grievous sins during Mass... and it was seen as more merciful to make confession a private matter between a priest who could speak for the whole community and the penitent.