I get exciting ideas about math a lot, and I was wondering if it is a bad idea to share them in question form to vet them, expand them, or show their lack of novelty. I know free exchange of ideas is good, but is it wise to exchange so broadly so early? If someone uses my idea from SE I know they are supposed to cite me, but could they take the career credit despite my having "The Idea"? Would I be a coauthor to the first paper from the idea?
There are several points that you should consider in this respect:
Most of the time our novel ideas are not as great as we think and there might be simple answers or counterexamples for what we have in mind. So it is better to think about a question in mathematics for a while and study some of the related literature, before sharing it with others. This has happened to me a lot, just see some of my questions in mathoverflow!
If you have an idea but you do not know how to tackle it, it is a sign that your idea is so raw and you don't own it yet. I experienced several times that a great idea is the one that comes with a clue for solution and extension. Otherwise, it is at most a spark in a well functioning brain of a hard working mathematician. I am being kind to mathematicians:-))
If your idea looks so great that can solve some big problems or answer some famous conjecture and you know how to pursue it, why do you want to share it? Just do it! And I believe such an idea deserves several years of works.
I suggest you ask related questions around your idea without giving a whole picture and what is your ultimate purpose. In this way not only you did not reveal it with every body, but you can also gather lots of valuable insights and knowledge to think about your main idea.
Read related questions and answers on mathoverflow or mathstackexchange. A great deal of wisdom are contained in these questions and answers and sometimes you get a clue about your idea.
Most mathematicians have bigger fish to fry. So, I do not worry to share some of my ideas with them, However, I do not think mathoverflow or mathstackexchange are good places for such interactions. Talking with mathematicians in conferences and/or even sending them emails look better procedures.