One of my tendencies as a GM, and even as a player, is to correct things when I think they're being stated wrong. This isn't malicious on my part. I just don't want people acting under the incorrect information, and it is very easy to get something wrong in an RPG. The problem with this though is that sometimes people are deliberately giving wrong information, and by chiming in you're disrupting the scene, the conversation, and making things more awkward.
There are a lot of reasons for someone to give the wrong info deliberately. Sometimes it is how their character saw it. Sometimes they feel their character just wouldn't get it or care enough to get it right. Sometimes they are actually trying to deceive and lie in character.
The reason doesn't matter. What matters is that when you chime in and bring things out of character to correct it, you can lose a lot of energy in the scene. In ensuring that the player knows what is up, and is saying it deliberately, you break them out of the moment. This can de-escalate desired in character tension.
Let Them Play
Because of this, of late I've started giving myself the following advice mid-session: let them play. Keep quiet, let them play the scene out IC, and don't interrupt either with clarifications or stuff going on in the world. Let the scene play out and see what happens.
Now if things slow down, or the world is primed to interrupt that is one thing, but 90% of the time when the PCs are having a conversation there is nothing immediately going on that could interrupt that. When the GM moves things along, they're literally moving things along either in time or location. So just let the players play out the scene.
That said, you still want to pay attention. If 2 of 6 players are doing a private scene, you can do stuff with the other 4 players. There is no reason that they can't also do something while those PCs are having their private scene. Even then, try to pay some attention to the conversation. Did something important get said wrong? if so, make a note for yourself. Were there questions they need answers for? Make a note.
Are things getting heated in character? Keep an eye on it. If it seems to be going out of character you can chime in with a "is everything cool, out of character I mean?"
Time For Correction
After the game make yourself available to the players on an out of character level for questions and clarifications. This is the time to ask about those things you made notes of. You can do it privately with the player, or openly at the table depending on how your group works. Mostly I don't keep secrets from the other players and trust them to keep their IC knowledge and OOC knowledge divided, so I can just ask. Other groups may do things differently, and that's just fine.
After the game is also a good time to make sure players are having fun out of character even if their characters are at each other's throat in character. Drama is fine. Tension is good. As long as both are kept in character. Out of character? It's a game. It's supposed to be fun.