St. Augustine, for example, presents this teaching as very much a matter of charitable obligation rather than ecclesial administration:
For our rebuke should be in love, not eager to wound, but anxious to amend. If you pass it by, you are become worse than he. He by doing you a wrong hath done himself a great hurt; you slight your brother's wound, and are more to blame for your silence than he for his ill words to you.
He goes on with a word of warning, that we don't always play the part of the wronged one:
And do you confess that by your sin against man you were lost; for if you were not lost, how has he gained you? Let none then make light of it when he sins against his brother.
St. John Chrysostom compares and contrasts the various ways a disciple of Jesus is to react when sinning or sinned against: the offender is to go to the offended (Mt 5:23) before offering his gifts at the altar; the trespassed against is to forgive the trespasser (Mt 6:12); the one sinned against is to go to the sinner (Mt 18:15). God doesn't care who started it, He wants His children to end it.
St. Jerome points out the custom of transferring this teaching somewhat:
If then your brother have sinned against you, or hurt you in any matter, you have power, indeed must needs forgive him, for we are charged to forgive our debtors their debts. But if a man sin against God, it is no longer in our decision. But we do all the contrary of this; where God is wronged we are merciful, where the affront is to ourselves we prosecute the quarrel.
That's not how I'd put it, at least in my case. I figure God can deal with His own affronts, but why else would He be so slow to reprove affronts to me if He didn't want me to prosecute them myself?