How Does Personal/Private Confession Work?

If we're talking about confession AND absolution then (strictly speaking) there is no such thing as personal/private confession and absolution.  That's because absolution is a word spoken to you from the outside.  You can't absolve yourself; it's God's Word spoken to you.

That said:  There is a Word of God that takes root in you, which you remember and respond to in faith later after you have heard it.  One of those Words, is the Word that announces God's forgiveness for Jesus' sake when you repent and turn from your sin.

So suppose you realize you have fallen into sin of one type or another.  (Maybe you were reading your morning devotions and it "called you out" on a sin you'd been ignoring or suppressing.)  Realizing that you have sinned once again, you drop to your knees before God saying, "God be merciful to me, a sinner!"

Now assuming you say this because you know God has already sent His Son Jesus to pay the price for all your sins (past, present and future) and promised to forgive your sins for Jesus' sake, then you are confessing to Him and "hearing" Him "speak" the words of absolution to you as a memory from an earlier sermon, Bible class or Christian conversation.

In almost every way, it looks like confession and absolution.  All it's missing is a second human being (aka the pastor) who hears the words you say and speaks the absolution back to you.

In view of this, many people ask: "Well, if I can 'get forgiven' without the embarrassment of anyone hearing about my sin, why would I ever confess to a pastor?"

Answer:  Because you might need to hear it directly.  Hear me now and believe me later:  There is something powerful about speaking your sins out loud to someone who (as a human) has no reason whatsoever to forgive you, only to hear them say the very words they have no reason to say (as a human).  This mirrors the relationship we have to God Himself:  Measured according to human standards of justice and law, there is no reason to expect His forgiveness.  Yet we plea for mercy and experience great relief (sometimes even surprise) when He grants it to us. The more ashamed or embarrassed you are to speak the sin aloud, the more profound the words of absolution will probably strike you.

I can't make you experience this for yourself, but I wish I could.  Once people have experienced genuine individual confession and absolution, they don't usually hesitate to go back again when they need it.  The rest are just missing out on a profound and very personal experience of God's grace.  They are no less saved, but they are missing a good gift that God means for them to have!