If you have a casual conversation with someone and it leads to a paper, what do you owe them? What if you had a casual conversation with someone, and that conversation lead to your findings?

What do you owe them?  When is it plagiarism?  I read somewhere that "first to submit, wins" - so basically the first to publish owns the idea.  Is this true?
The problem with this question is that it is way too vague. Suppose you have a casual conversation, and somebody says "I've been looking at Trudeau gadgets, and I think I can show that they all have a canonical commuting umbrella." If you then publish a paper showing that all Trudeau gadgets have a canonical commuting parasol, there is no question that this is unethical, even if you didn't hear all the details of his proof. (Things like this have happened occasionally, and sometimes the unethical author gets away with it, but I'm not sure doing this is good for your career ... people will probably hear about it). 

However, suppose you are having a casual conversation about your research, and you explain to somebody what a Trudeau gadget is, and she says "That reminds me of Smith's umbrella theorem." You then look up Smith's umbrella theorem, and it is exactly what you need to prove your result. Here, it would be nice to thank her in the acknowledgments of your paper, but you certainly don't need to make her a coauthor.

So there is really no way to answer this question without knowing more details.