We used to leave the room back in the 90s, or we would have players close their eyes, cover their ears and sing a song when the GM was telling someone a secret. Nowadays we're much too lazy for all that. We use the occasional written note, or just nod and give thumbs up to signal agreement between player and GM.
It's not just laziness, it's actually a lot of fun for the players to know more than the characters. Consider a suspenseful movie scene. If we know less than the character, we try to understand what's going on and frantically try to make sense of what's going on. If we know exactly what the character knows, it can be immersive. If we know more than the character, it's suspenseful. If the character enters a room and we know there's a killer hiding behind the door, it's more suspenseful than if we are as surprised as the character when the killer emerges.
I know that some players put immersion as the single highest goal of any RPG session, but I don't. I much prefer to be a simultaneous participant in and audience of the story.
(Obvious spoilers for the scenario now.)
In Chariots, I was the GM with three players experienced in other games. The hidden android player didn't get any extra instructions, I just told him "yes, you have the correct agenda" because the name on top was different. After each act, I collected the agendas and just told the players one by one without disclosing the agenda: "You get a story point... *turns to next player* hmm, you don't", and so on.
They were a bit suspicious of each other, but in act 1, the agendas are not in conflict, so they got along well. In act 2, the Weyland-Yutani rep PC was killed, and the player decided to take over the Weyland-Yutani rep NPC. They had already found the lab with the vials of black liquid and had assembled in the cockpit of the Cronus. As soon as he read his new agenda, the player asked "No one's paying any attention to me, right? I leave for the med bay and take on of the black liquid vials!" The other players basically cheered, because it was such a cinematic thing, especially as she was ambushed by a neomorph in the same scene and killed in one round. It was like taken from a movie: she betrayed the group and was immediately punished for it.
The other real betrayal was when Lucas the android revealed himself. He walked off with the doctor NPC (now played by the Weyland-Yutani player mentioned before) and the android Ava and shot them both with a shotgun. Since this was PvP I should have taken control of his character after the scene, but I'm so used to games where players can have their separate stories that I forgot. He continued to find the other PC and shot him too. He then steered the ship towards human space, leaving the two NPCs doing a spacewalk to repair the ship stranded on the outside!
My point is that this was all in the open, at least after things had been revealed. The Lucas player said afterwards that he had some questions during the session on what he could or could not do to sabotage the mission, but he just kept playing to the best of his ability and it all turned out great. If the players *really* had wanted to be secretive, I would have made them pass me a written note, and if possible, I would have answered "Yes" or "No".
I think your suspicion that leaving the room will slow the pace and remove tension is entirely correct, but I think it's possible to avoid by note passing or doing things in the open. Players can usually hide any initial maneuverings from each other, but after a while it just turns annoying and I'd much rather have it in the open.