Side comment:
There seems to be an argument, directly or indirectly by multiple posters, that a player needs to "pay" in some way for having a character with a dump stat and part of paying for it is, for example, folding in an argument between characters, acting stupid or playing dumb, not sussing out a lie, etc. Or abiding by the result of an ability check by another player's character.
I think when we're saying "pay" we're saying "play the character's faults" or "play the character as written". If you've got a dumb or weak willed character who suddenly sprouts iron-willed resistance to a charming character's arguments....well this seems the same as if the physically weak character simply declares that he resists the grapple of the stronger character or something like that. That is to say, if we would resolve the physical conflict with dice, then we should resolve the social conflict with dice as well. (D&D may not be particularly suited to this sort of thing, but that's a different discussion.*)
The potential conflict is in a perceived loss of player agency on the part of the weak-willed character and a de-protagonization of the charming character. Some of mentioned that they just remove the part of player agency that would allow for PvP die-rolling entirely. To me, that seems an incomplete and somewhat heavy-handed, but that can vary from table to table.
This can be problematic in at least one way when a player attempts to make up for his character's written stats with their own Intelligence or Charisma. That is, the 8 Charisma PC who nonetheless fast-talks his way past the guards because the player fast-talked the DM, or the 8 Int PC solves the complex mathematical cypher puzzle using his player's physics degree. I'm not sure I'd call it cheating so much as unsporting.
I therefore submit that the player is not really gaining any benefit from the DM having the players work out their PvP issues between them without reference to the mechanics. I'd even go so far to say that, if anything, doing so is a penalty of sorts, a tax on the player for their build choices which is unnecessary. There is, after all, a whole world of villains and monsters out there to lie to, steal from, and murder and all manner of social and exploration challenges in between. Why anyone would turn to the party for that conflict is a little baffling. But in any case those dump stats will come back to haunt that character without any additional burden of mechanics being used in PvP.
I hope that made sense. I'm still in the process of thinking it through.
Does character A talking to character B to try to convince them of a course of action count as conflict? I mean, IRL and in the genre sources, characters will discuss things or argue with each other a lot. I don't think we're talking about something like "give me your wallet", but more like "let's go defend the village before we go hunting the orcs." (Then again, its not like picking your friend's pocket is unheard of in the relevant media, either.)
The rest of what your talking about will vary quite a bit from table to table. Some groups start and end at the dungeon entrance while others spend tons of time politicking around town. Part of being a good GM would be taking a clue from the players' choices and characters' stats and tailoring adventures so that folks get a reasonable chance to shine. However, we are not all blessed with such DMs. ::shrug::
I don't know that there is a universally applicable answer to this question. However, I have seen far more (especially old-school, it would seem) folks who would forbid the social skill checks for "Can I try to Persuade him that we should protect the village?" but cheerfully let the dice roll for "Enough! I try to cut the thief down for his insolence." and that, IMO, leads to a broken game.
*One way is that D&D tends to resolve social situations in very few die rolls and mechanical interactions, whereas combat is typically resolved with many. Which, come to think of it, can lead to the odd situation where a 30 second combat might take 30 minutes to resolve but a 30 minute argument might take 30 seconds to resolve.