There are potential arguments to be had when it comes to characters persuading Non-Player Characters, and I think they come down to Player Skill Vs Character Skill.

Larry the player doesn't necessarily have the skills to make an ultra-persuasive argument, but Larry the bard definitely does.

Does Larry the player need to be able to present their argument skillfully, or does Larry simply say what he wants to argue, and roll to see how well it goes?

I've heard of game masters handling this in different ways, such as having the difficulty of the check be affected by how well the player makes their case, or players getting a bonus to their roll if they act exceptionally well.

Or, of course, just let player making clear what their character wants to argue, and rolling to see how well they do.

It's one of those things that I assume boils down to the players, and what the group finds the most fun.

I'm also assuming there are rules on how to handle such situations, but it's also my understanding that groups will wind up having their own rules for such things.