The sort of play I’d be going for with Truths would be one in which character secrets are known to everyone on the player level, what I think of as narrative secrets. The purpose and role of a narrative secret is very different from what I think of as a competitive secret. In a PvP competitive game, secrets are generally things that could be used against you that you’re trying to keep secret at all costs, for both mechanical benefit and being-in-character reasons.
In a competitive game, player secrets are often an important part of gameplay. PvP success often involves outsmarting, outmaneuvering, and/or tricking your opposition, and secrets are a big part of this. (LARPs like this can even be called "secrets and powers" LARPs.) Whether it’s a backstory thing like “the prime minister is having an affair” or a tactical thing like “those three have formed a secret group and are going to try to assassinate the prime minister tonight”, the keeping of and betrayal of secrets can be the difference between success and failure, and especially when you don’t know who your opposition is keeping secrets on a player level is important for success. Even if “metagaming” is forbidden, it might be hard for a player to not let a secret they know influence their play. (The main exception I see in practice, where players are let in on stuff their character shouldn't know, is when someone is knocked unconscious and the player ends up overhearing something. This is one of several reasons why I’m not into unconsciousness in secrets-and-powers LARPs.)
In cooperative games, on the other hand, things are different. Sometimes players like keeping player-level secrets even in cooperative games. This can lead to a certain shock factor if done well/with intention, but I mostly see this as a blind adherence to the idea that “metagaming is bad”, which is not something I’m into. On the other hand, if players know about secrets, they can bring them into play in interesting ways, coming up with things their characters might say or notice that might lead to cool scenes because the secret is being hinted at, similarly to how in a TV show or book there might be foreshadowing of a secret or behavior that seems strange but later makes sense once a secret comes out. This can work very well in tabletopping, as well as more narrativist LARPs (see the Jeepform principle of transparency and "steering" as positive metagaming). Even if you’re not going to that level of player coordination, in games that aren’t about keeping secrets at all costs it can lead to more interesting play if players let their secrets come out during game, and thus letting other characters react, rather than sitting on them until game end.
Given that players may have different backgrounds or expectations about secrets, it’s often a good idea to explicitly work to get people on the same page. People used to serious competitive LARPs may keep secrets until game end even in a lighthearted LARP where it’d lead to more interesting play if they came out dramatically; conversely, players who are used to playing up secrets in roleplaying may be unhappy if they unexpectedly get killed because they let a secret slip. Providing expectations like this in your rules document, alongside your mechanics and genre expectations, can help avoid bad experiences. (Another option I used in a mostly-competitive game was to have revealing the secret, in the right way, be part of the plot, which set up a good scene.)
In general, it's good to make sure you know what purpose secrets are serving in your game, make sure the way you treat them is serving this purpose, and communicate about this to get people on the same page.