Example, if you run a marketing agency and your competitor also runs a marketing agency, should you be sharing samples of your work?
As a marketing agency, your work is designed to reach people and be publicly viewable anyway, right? So I don't see the problem. Why not share specimens that have already been shared with the world at large?
Today's competitor may be tomorrow's coworker or collaborator. Perhaps your peer might wish to hire you or bring you on as a partner some day ... or vice versa. Maybe the overlap between your services and skill sets is only partial. In some cases, I and my competitors team up to work with a single client; so we actually refer leads to one another.
If you're aloof and standoffish, then you may be closing doors to future opportunities. Since the risk of being open appears minimal, just relax and talk about the best work you do.
Well it can be socially awkward to say, "no, I'm sorry", and it can be pretty much impossible for some people to say that. The fact that you're asking the questions means you're looking for a way to not say that.
So instead, just show them some of your older and / or lower quality work. That way there's no bad blood, and you're only showing stuff that's old news, and you are also convincing them that you're doing lower quality work than you actually are, which will make them be more comfortable and therefore easier to out-compete, if you're out do do that.