There's a sort of sense of involvement in a thread you started. and maybe you make a bit more effort to nudge it in the direction you want and to refresh it if it's fading prematurely. But it's not a business of possessiion or having any rights to it. Essentially you let it go and see how it flies.

I mean if you were in a pub or somewhere and you'd started talking about something, and the conversation moved off onto something else, you wouldn't be saying "Here, this is my conversation, don't you go hijacking it".

You might say something like "To get back to what we were talking about, here's how I see it" - within limits. But you might say the same in a conversation that had started before you joined in. And you'd be taken aback if one of the the others said "No, this is my conversation, I decide what we're talking about."