Released offenses

John 20:23 NAS
"If you forgive the sins of any, their sins have been forgiven them; if you retain the sins of any, they have been retained."

The word for "sin" here is hamartia, and can be translated as an offense.
The word for "forgive" is aphiemi. This can be translated as forgive, or other words like remit. Remit is a word we don't use much, but means to pardon, to refrain from enforcing a punishment.
The word for "retained" is krateo. Retained is one of the literal translations so lets look at the definition of "retain". One definition is to keep in mind, to remember, or to hold in place, like a retaining wall.

Paraphrased this could say: If you refrain from punishing the offenses of any, their offenses are pardoned; if you hold in place the offenses of any, you retain them in yourself.

I believe that this verse is not telling us that we have the power to forgive sins against God, but that we have the power to forgive offenses toward ourselves. If we forgive someone who offends us, we don't enforce the punishment, even if deserved.

The second half of the verse does not talk about forgiving, it talks about retaining. It also is talking about what we retain, not what the other person retains. I believe what this is telling us is that when we don't forgive, we retain. Those offenses are kept in us, building up until let go, until the retaining wall is torn down, releasing the offenses and the offenders.

Forgiveness is as much or more for the offendee's benefit as it is for the offender's.

The Root Produces the Fruit

John 15
⁴ Remain in me, and I will remain in you. For a branch cannot produce fruit if it is severed from the vine, and you cannot be fruitful unless you remain in me.
⁵ “Yes, I am the vine; you are the branches. Those who remain in me, and I in them, will produce much fruit.
¹⁰ When you obey my commandments, you remain in my love
¹² This is my commandment: Love each other in the same way I have loved you.

Summed up;
We bear fruit by being joined to the vine (Jesus).
We are joined to the vine by remaining in Him.
We remain in Him by keeping His commands.
His root commandment is to love.

Therefore, we bear fruit by loving.