'Owning it' is a very big thing.
When people make a mistake, we just want them to own it. Admit it. Be the center of some public shaming for a while. Let us point and whisper; take the attention off of us. Give us a chance to shove those skeletons back into our own closet.
Own it, wear it, spend some time feeling guilty about it.
But, what's the statute of limitations on a screw-up? How long, exactly, must one wear that scarlet letter? Until the jokes run dry? Until the next person screws up?
Discussing other people's imperfections is rarely a slow conversation, because it's a whole lot easier than looking in the mirror to find our own.
Owning it is important. You can admit what you've done, make up a little story about how it has changed you, and move on.
But for the love of God, do not let it define you.