Prop bets, going south?

So I was just watching a recent stream of live poker and saw something very peculiar. Before I mentioned who, what, when and where, I wanted to ask people that are more familiar with running prop bets while they play live a question.

If you are playing props between you and another player at the table and you are using the chips you have in play to bet the person with. If you win are those chips supposed to remain in play?

I have not done that but as i have seen in "hight stakes poker" for example the prop bets were running on separate sheet of paper and settled outside of the poker game and not with the chips which are on the table.