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.
Apology. Never have casual conversations with people. Its rude.
Also what Peter Shor said.
I would also add that if you ask the question, then usually the right thing is to offer this person to be a coauthor. If the person is decent and it was really casual they would thank you, ask you to put them in the ack, and decline to be a coauthor. If it was really minor and they still stay as coauthors, they would feel thankful to you. This might be beneficial later on - they might work hard on the paper and push it forward, you have this person as a coauthor (beneficial if they are well known), they would feel they "owe" you, etc.
The other possibility is that this person is a free rider and becomes a coauthor without doing anything and not having contributed anything. This is useful thing to know - you carefully avoid working/talking with this person. You gained useful information for little cost.
And this is all to avoid the worst case - you do not offer this person to be a coauthor and they become bitter and clinging to their guns and bibles (not necessarily in this order). They start spreading stories about you that you had stole their excellent paper. They shoot down your papers whenever they can (referee reports, PCs, etc). As your career collapse you become bitter and clinging to your gun - and you move to Texas. And all of this just because you did not want to offer them a coauthorship - sounds like a bad deal to me.
In short, there is enough glory for everybody - always mistake on the side of sharing glory... Publishing papers is not a zero sum game - there are long terms considerations that arise out of the fact that this a repeated game with side effects.
