How do people who believe in the act of confession understand 1 John 1:9?

repentance → confession → absolution (receiving) → reconciliation

To put confession of sin in greater perspective, let's remember that all denominations, even other religions as well, recommend confession of sin. In Judaism, teshuvah is central, and the meaning of repentance as a "return to God", of "turning from the wrong path to the right path" is shared with Christianity.

Like the father in the parable of the prodigal son (Luke 15:11-32), God has already forgiven us in Christ even before confession (v.20) just when the father first notices the first sign of our repentant heart (vv. 17-19). But to fully appropriate His forgiveness and making the benefits of forgiveness "real" to us we need to walk back into His presence like the prodigal son (v.20), then confess verbally (v. 21), then hearing forgiveness to fully receive the fact that the other party has forgiven us (v. 22-24). As the parable illustrates, this act of confession of sin is tightly coupled to the receiving of forgiveness. The celebration that the father gives to the prodigal son is a further display of the enduring reconciliation as the party goes into the night, making it supremely real.

Jesus very much emphasizes reconciliation whether it is our fault or not that in the Sermon on the Mount he mentioned this in at least 4 places that are textually tightly connected and can be seen as 4 steps of increasing difficulty:

1. When we worship God, to stop and be reconciled FIRST (Matt 5:23-24)

   The text does not say clearly why our "brother has something against" us. We may be at fault or we are not. What is clear, "be reconciled" is the focus. Notice too, that "brother" implies someone we fellowship with, not a stranger, not an enemy. That makes this the easiest command to obey.

2. In the teaching about retaliation (Matt 5:38-42)

   In the next teaching we are clearly not at fault. Yet how can we perform the command "do not resist" when "the one who is evil" (not a brother) has already done us harm, without at least forgiving that person in the manner that Christ did on the cross saying "Father, forgive them, for they don't know what they are doing?" (Luke 23:34).

3. In the teaching about love for enemies (Matt 5:43-48)

   In the next teaching Jesus takes it one step further! Not only should we NOT retaliate on "the one who is evil" (which now becomes our enemy), we are asked to love them too! The reasoning given is that so we can be "sons of our Father" as the "heavenly Father is perfect". Clearly, upfront forgiveness is necessary which is reconciliation in our heart, a readiness to offer it when our enemy confess to us someday.

4. In the teaching about prayer (Matt 6:5-15)

   As if it's not enough, Jesus takes it another step further!! The Lord's prayer says: "forgive us our debts, as we also have forgiven our debtors" (v. 12) and immediately added "for if you forgive others their trespasses, your heavenly Father will also forgive you" (v. 14) and added a warning that our own forgiveness by our Father is conditioned by "forgive others their trespasses" (v. 15). Now, not only we need to love our enemies by forgiving them, our own reconciliation with God is at stake too!

Loving God as readiness to reconcile with everyone

To love God means to want and to do what God wants and does. This means our will needs to be 100% in harmony with God's so we can will and do what is good for all people around us, including our enemies. This is hard. What makes it easier is for our heart's desires to ALSO be 100% in harmony with God's desires so we can easily want what God wants, which lead naturally to acts of loving kindness. This is what will happen in heaven, where the desire to sin doesn't even exist since our desires is in perfect harmony with God's desires.

What God demonstrated in the Parable of the prodigal son is that his heart is always in a state of reconciliation with us and he is always waiting for us to walk back, ready to run to embrace us at the first sign of repentance even before we confess. He then celebrates the reconciliation. Jesus said many other parables to the same effect, like the Parable of the lost lamb or Parable of the lost coin. If what God desires of his enemies is reconciliation, then if we love God we need to do the same. That means doing the 4 commands of Christ described above. This is hard.

Absolution as medicine for our heart

To state your question more concisely, you're basically asking why absolution is necessary when 1 John 1:9 simply says that God will "forgive us our sins" "when we confess our sins"? In other words, if we have already done the steps of "walking back in repentance" and did the "confession" (probably in prayer), why do we need priests to perform the rite of absolution? As the previous section shows, even Jesus has already forgiven us, maybe we don't even need to confess?

I suggest that the practice of absolution is to complete the process of reconciliation, which is the larger reason why confession of sin is needed in the first place. Absolution can be conceived as "receiving forgiveness through sight and sound" followed by a "celebration of reconciliation". The whole experience of joyful reconciliation with God should make it easier for us to experience God's forgiveness in our heart. We can then have the desires necessary for having our heart ready to reconcile with others just like the father of the prodigal son.

We need to bring into view here another verse, James 5:16:

Therefore, confess your sins to one another and pray for one another, that you may be healed. The prayer of a righteous person has great power as it is working.

which adds 2 elements:

• the prayer of another person
• healing

With James 5:16 we can put absolution in another way: to bring healing to our heart so our heart can also have God's desires to reconcile with others.

Absolution is practiced by many Christian denominations

Hearing the words of absolution from a priest is like hearing the words of forgiveness from Christ himself, because when a priest is doing the rite of absolution he is doing it in persona Christi. It is true that the Catholic Church invests additional benefit and requirement on the Sacrament of Reconciliation but for this answer I'm focusing on the commonality with Eastern Orthodox, Lutheran (Holy Absolution), and the Anglican churches (Confession and absolution).

Even in churches that do not do priestly absolution (such as the Reformed churches), there is a place in the liturgy for hearing forgiveness. In the Presbyterian liturgy, there is the "Call to Confession" where 1 John 1:8-9 is read, followed by the "Confession of Sin" where a prayer of confession may be said aloud (or in silence) by the congregation. Then the minister will say the following aloud in the "Declaration of Forgiveness" as liturgical option #1:

The mercy of the Lord is from everlasting to everlasting.
I declare to you, in the name of Jesus Christ, you are forgiven.

May the God of mercy, who forgives you all your sins, strengthen you in all goodness, and by the power of the Holy Spirit keep you in eternal life.

(source: pages 52-56 of Book of Common Worship).

"confess" in 1 John 1:9 as "declaring something to another human being"

In a March 2021 Catholic Answers article God Wants You to Confess Your Sins to a Priest with the subtitle "Why confess your sins to a priest when you can just confess them straight to God?" Catholic apologist Trent Horn suggests that the primary meaning of "confess" for 1 John 1:9 is suggested by 1 John 2:23 which implies public confession. He then brought into view other NT verses containing the same Greek verb h*m*logeō which also implies public confession (Acts 7:17, Matt 7:23, John 1:20, John 12:42, Matt 10:32, Heb 11:13, etc.) as well as Didache which gave believers the following instruction:

in your gatherings, confess your transgressions, and do not come for prayer with a guilty conscience

Then there is also public confession of sin in the Old Testament (Lev 5:5-6, Prov 28:13, Sir 4:25-26, Dan 9:20) noted by Fr. Raymond Brown who wrote the Anchor Bible commentary on 1 John.

Does it mean that if we don't confess our sin publicly or to another human being we will not be forgiven? That is the wrong conclusion, since CCC 1458 taught we should confess to God directly whenever we feel guilty with an act of perfect contrition. And it is still only God who forgives, even though God has chosen to use human intermediaries. But the norm is through confessing our sins aloud (auricular confession) to one another (James 5:16) as soon as possible (Code of Canon Law 916).

Conclusion

What is common to all denominations that practice confession of sin mentioned above is that absolution is the concrete receiving of forgiveness to make REAL our reconciliation experience with God, giving us healing to our sinful tendency, which in turn help our hearts to have the desires to offer reconciliation to others as Jesus has commanded.

Also, as Trent Horn's answer shows 1 John 1:9

does not refute the concept of sacramental confession, because this verse does not say we should confess our sins only in private prayer addressed to God. First John 1:9 refers to the practice of “confessing our sins”—it does not say to whom or how we should make our confessions.