The person laughing at their own mistake still deserves a choice about who repeats it.

Laughing at yourself is a way of setting the terms: you decide the audience, the timing, and how much of the story gets told. When somebody else carries that story onward, the terms change hands. What was a moment of ease becomes a thing you have to hear about from people you never told.

Before repeating a funny story that belongs to someone else, consider asking whether they mind. Their willingness to laugh in one room is not permission for every other room.