Methinks you are bringing pre-5e style to your stance.
5e might just work best when the DM calls for the rolls, not the players. It's right there in the "How to Play" section of the PHB. YMMV of course
Let me ask this from the other side. One that doesn't affect character agency.
Character A tells the group something. The character is lying.
Character B is suspicious - maybe they think character A is charmed, or carrying out a mission for their faction, or it doesn't jive with other information. Player B asks the DM if their character thinks Character A is lying.
Is it in-line for the DM to ask for a bluff vs. insight role if the first character wasn't telling the truth?
(Alternately: for those who are using the Isereth (?) method where pvp is determined by the person being affected, is the bluff a "pvp attack" and it needs to be disclosed to player B that it was a lie so they can determine how they respond?)
This is just wondering about social skill use vs. other characters when player agency isn't at risk. I think the answer to this sort might help define where the line is for some.
Instead of playing word games, answer the actual question. I tried to phrase it in such such a way that those who "don't let players call for rolls" would accept it, and now you're dodging the question with fancy wordplay. This does not add to the conversation.
Character B is attempting to figure out if "A" is lying. Regardless if "A" is a PC or NPC. There happens to be a useful skill called Insight to help tell this and there's another skill called Bluff to tell the lie in a believable way.
I fully believe you are capable of putting together some sort of use of these skills and a potential lie that can answer the question instead of dodging ti with a snarky answer that blocks the conversation from happening.
If a player was unsure or in the dark if another player was having their character lie, would you require the player be informed, or does this play out in-game at the character level? However you envision that occuring.
Saying that all players automatically recognize all lies may be a true at your table but isn't particularly useful in the realm of "all tables" for furthering the discussion. Especially AL games where people may not be there week to week or may join a game already in progress, where factions give out secret missions, etc.
Let me ask this from the other side. One that doesn't affect character agency.
Character A tells the group something. The character is lying.
Character B is suspicious - maybe they think character A is charmed, or carrying out a mission for their faction, or it doesn't jive with other information. Player B asks the DM if their character thinks Character A is lying.
Is it in-line for the DM to ask for a bluff vs. insight role if the first character wasn't telling the truth?
(Alternately: for those who are using the Isereth (?) method where pvp is determined by the person being affected, is the bluff a "pvp attack" and it needs to be disclosed to player B that it was a lie so they can determine how they respond?)
This is just wondering about social skill use vs. other characters when player agency isn't at risk. I think the answer to this sort might help define where the line is for some.
(By the way to get iserith to give you the answer to your question you desire you are going to have to drop the "what is my pc thinking line" because he would never even allow that in his game. Instead translate that to iserith playspeak, "my PC tries to determine if the other PC is lying". Ask iserith what he would do here and I bet he would call for a roll as described. It doesn't present quite the catch 22 that you thought it would in that scenario though.)
[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.
Welp I lost that bet lol. But that's influenced from your way of resolving pc vs pc conflict.
However, perhaps more importantly for everyone to understand is that even if it was an npc in question instead of another pc you still wouldn't allow the "hey DM does my pc think they are lying" comment. You would have them say what their PC was trying to do as an action, "my pc is trying to determine if NPC is lying". Then if you determined it uncertain you would call for the roll.