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You’re getting dragged, but I think I get your point: that it might seed suspicion and distrust among not only the characters but the players?
I think it should be Rule 0 that the DM and players all respect each other’s boundaries, so consent is always part of the equation IMHO. Likewise, if everyone at the table insists on playing a certain way that I’m uncomfortable with, then it’s time for me to find a new table.
Oh agreed: I just thought the meta-nature of the players themselves being unsure of the veracity of character information might make things weird for folks uncomfortable with gossip. In any case I think it sounds like a fun idea!