Ovinomancer
No flips for you!
[MENTION=16814]Ovinomancer[/MENTION] has it. It's just worked out by the players involved in a way that they believe will be fun for everyone and that contributes to an exciting, memorable story. I see no need for mechanics here.
For transparency, a few years back I got into an argument with you about using social mechanics against players. I was on the other side, then. I understand the intent behind both sides, and I do play some games that do allow this kind of PVP social mechanics being binding. 5e isn't one of those games, though. There's no inherent penalty for failure, for instance. If you try to persuade the other PC, there's not "if I fail, they persuade me" mechanic. That, as you well note usually, means you shouldn't roll -- if there's no penalty for failure, the roll isn't interesting enough to make. You could shoehorn it in, but, frankly, the nature of 5e skill math is such that it's not a good fit, and you'll find players aren't really willing to risk the all-or-nothing effects of such things. This first 1 on a d20 from the super-convincing bard vs the barbarian meaning the bard was convinced by the barbarian and must abide by it and you'll see grumbling about stupid mechanics. Games that do PVP well almost always use generic and well balanced mechanics that always allow for a moderate chance of failure (at least) and the challenge is really to see who gets their way in the fiction rather than 100% win/lose. The fiction moves forward either way.