Tom, something similar happens in a song circle I attend in Portland, OR. Most songs we sing have been done at varying tempos, in various keys and with various styles, but a few folks in the group will often request a particular song repeatedly and a sort of >group norm tends to develop as to how to do that song.
Sometimes that group norm is quite different from the best known version of the song. And often the group rhythm lacks coordination and definition (because the person leading it is inconsistent or because there are "too many cooks".*
It's not so much that any one person owns the song, but if someone requests the song and then leads it in a tempo, key, style, or rhythm different from the way it's been done in the past, there is often resistance on the part of some people, even if quite a few people seem to prefer the "new" version.

When it comes to
performance (as opposed to jams and song circles), I personally have no qualms about doing the same songs other people do--unless our versions are pretty much the same and they do the song better than I do. In one club, I've a couple of times started into a song and had another musician call out, with (mostly) mock chagrin, "Hey, that's my song!" (or something like that) Then I've invited them to join me on stage, and we both (all) jammed on that song. This is great when the people joining you are good harmonizers/jammers and don't try to override your style.

Liz, "Dueling Kazoos!" Love it!

Greg, Re there being "enough [songs] for all to have a few of their "own" if they want to," I tend to agree, with one exception. Sometimes I've kind of made a song "mine," (worked up my own version that I really like, play very well, and have been performing for a long time) before I start going to a particular performance venue or jam/song circle. If someone else there also has made it "theirs," I see no reason why I should have to abandon the song for that venue. (We just probably won't both do the song the same night.) I don't see this as encroaching on the territory of someone who already had it as one of their special songs before I 'adopted' it.

Genie