From another angle: is it a built-in table expectation that your PC can always trust someone else's PC? (and if so, why?)
To flip that last one around: are PCs allowed to be untrustworthy with regards to other PCs but not to the extent of outright PvP? Can a PC be a spy for the party's enemy, for instance?
we go back and forth on this.
right now we have talked about the pattern we have noticed... We go through a stage where we basicly force all players to be on the same page with there characters... then we get board, all agree we are mature enough to go against it... then we open it up and push that envelope over and over again then something goes wrong and derails a game... and then we lock it down and the pattern repeats.
some noticeable examples: in 2e we had a player who was 100% for pvp and pvdm he tried to 'lay traps' even for the DM... we got to the point where we stopped playing with him.
in 3e especially early on we all made different characters with different goals and sometimes split the party. Until Paul (yes his real name) who would brag about "I have the avoid DM plot hook feat in real life" and would go out of his way to try to derail games... and he did a few times. We then made the rule "This is the game concept. If you want to have your character go off on there own and do something without everyone cool... they are now an NPC and you can make a new character" but then things got better and problems went away and we played an evil game and an any alignment game and things went AWESOME... then came the down side.
we had a game set in a 3pp setting, and we all made characters with our own objectives... but we ended up finding out about these swords of power and this mind flayer invasion... we figured out that 2 of us had very oppsed goals though, and we had to agree to (in game) put that aside it twice almost split the party in half. It tokk the DM asking us to please trust each other to get it going as long as it did. but around level 7 (when the DM told us he planned up to epic levels) the game blew up when one player stole the 3 swords we had, and took them to his king. The DM tried to get the game back on the table but that player was not trusted... not his character for sure (and he wanted to bring the character back) and I personally said I didn't even fully trust the DM... he knew that player was planning on betraying us and still told us to trust each other.... so we went back to good/neutral align only and have to work together.
Right before we split off D&D for a while I thought things were cool and opened up ideas and we played in my world... 1 player (not either of the above but someone who was no problem in those) got his hands on a hand out... that spelled out alot of what was going on. But he read it outloud but editied the hand out. (this was game 4) we played for months and the player kept the hand out in his character foler with his other notes. He also forgot (in and out of game) that he didn't share the info. So something happened that if they knew what that hand out said would be perfectly logical... but 2 players (Out of game) where mad about the twist that had not been foreshadowed and if anything (like above) felt the DM (me in this case) cheated by putting his thumb out of game on things... but I broke and said "XXXX knew for months" and that player forgot had to go back reread the hand out then say "opps... I wanted to read that book and I may have screwed up the game..."
(FYI that game did not end we did get it back on track but with issues)
about a year or year and a half we stopped playing D&D. we didin't like the 3e system and went away from it but we found a different idea. (while playing Mage we figured out how to make a rule/concept for Vampire) "What if we were all the actors AND the audiance"
So that is what we have done coming back for 4e then slowly moving to 5e. We allow PCs to go off on there own, and other players just take up NPC roles with them, then that player takes an NPC role with the main cast. No notes passed no secrets kept. If someone is secretly working against the other characters... out of game everyone at the table knows and is laughing cheering and going on. We also all have red cards. Anyone (DM, Player anyone) can just say "NO" to a twist to an idea, to anything. by red carding (orginally just a word veto then literally Becky made index cards with red duct tape and an X on one side and a circle with a line through on the other, but now on Roll20 just words again even though I have the cards here)
We are now talking about running curse of strahd with some twists... and maybe going back to keeping (some) secrets but we aren't sure...
edit: my biggest complaint with splitting the party is lack of fun for people who are not there in game... keeping everyone engaged is the most important part now to me.