I have a Deaf sibling, and spent much of my time growing up around d/Deaf individuals.
There are absolutely different social boundaries and rules surrounding sign language conversations. It is actually both common and acceptable for an individual to simply enter a conversation that in hearing culture would be considered "private" in a way that, again, hearing culture would see as "intrusive" or "eavesdropping." It is not considered to be either in sign language conversations - in fact, it's expected behavior. As a hearing person I do find it to be uncomfortable, and strange, but that's me being hearing. I would never call someone out on that behavior, as then I would be the one acting strangely.
I think it has a lot to do with the natural differences between the languages - it is simply not possible to have a spoken conversation with five people across a gymnasium-size space. It's entirely possible, even normal, to do so in sign language (I have in fact been hit on across such a space while my sibling was visiting colleges. That was very strange indeed.)
Now there are
"rules" about privacy, but they are different than most of the posters here are expecting. Two individuals having a conversation in a corner, facing each other and backs to the group, signing close to the body - THAT is a private conversation and you are not supposed to intrude. Watching a conversation due to nosiness/curiosity and not intending (or being able to) engage, THAT is rude. Two people talking to each other in the open in a farm? Clearly exchanging signs in what appears to be a teaching/practicing exercise? Nope, totally okay to enter that interaction - in fact, expected, especially if an error comes up.
As for the questions about "correcting" and is that okay given the variation in signs - yes, it is okay, expected, and in some cases very, VERY important. I moved across country in adulthood and was chatting with a Deaf colleague who looked horrified and quickly corrected my sign for "work." In that area (within the same country I grew up in), my sign for "work" was....um, the sign for scrabble-related activities. My sign for the...activities....in question involved a similar wrist motion but different finger placements. The idea is to learn the signs that are appropriate for the area you are in, so that you don't wind up telling people that you are going to "Work" when you mean "work"
Lastly, in thinking about this I do have to say that I think it would be likely that I might have this type of interaction even in a spoken language. If I noticed someone nearby who was clearly not a native speaker of English, and teaching his/her child the English word for an animal but making an error in pronunciation (not just an accent or something) I would likely offer the correct pronunciation. Just because...well, I know it, and they clearly seem to want to know it, so why not help out? I don't think I'd be more or less likely to do this with a man than a woman, though I'd be more likely to do it if their pronunciation was skating close to a word with a different meaning.