My barn manager/owner will only allow riding instructors to teach there if they have insurance. If they don't, she will offer to put them on hers, but it costs them something. She's really the first person I've ever encountered that has this rule, but now that I know about it, I can't see why other people don't either! The barn owner could be sued just as easily for not demanding insurance from the instructor. Of course, some instructor's don't make enough from lessons to warrant the expenditure of insurance. It's a mixed pot.
If you're worried about people making trouble, why not just show your proof of insurance to the barn owner, and leave it at that? You could always tell people if they ask, but it doesn't make you vunerable in any way.