Sometimes someone reveals a piece information that is significant to other people, and the other people immediately want to discuss it, but they can't do so without alerting the first person that what they said is important. And if they don't want that person to know that they let slip something important, then they have to find another time or place or method to talk about it.
Players in a game have the handy method of discussing things out of character. In the normal course of gaming events this is not a problem. But when revelation and discussion of information is a significant plot point in your story, you may want to be strict on disallowing out-of-character discussion in front of NPCs (or in this case, other PCs).
Players will naturally want to drop out of character as soon as they hear something intriguing and worthy of discussion. You need to be ready to intercept this and let your players know - gently - that they can't discuss or share information without the person in front of them overhearing. Then just leave it up to them what to do about it. You've given the warning, so if they do discuss things, then let the relevant character overhear it and react accordingly. But generally players will take the hint and try to get out of the current encounter so they discuss things more freely - which can be a fine roleplaying exercise, even tricky if you play it right.
Commentary by Keybounce (who has not seen the movie)
Well, he almost said exactly what I thought he was going to say. And sure enough, Rey thinks that there is something wrong with this story. Of course, the way he said it doesn't sound quite so outlandish as the way I said it. I give him points for honesty, points for completeness; I take away points for probably spilling more beans than he should spill.
And we have an out-of-character check to make sure that Player One understood what Player Two said. It does look like Rey is calming down; no longer about to kill Finn for wearing Poe's jacket
Commentary by memnarch (who has not seen the movie)
Annie did very well in describing the idea Team Evil had. Not too much detail, nothing extremely out of the ordinary, waiting to be asked to provide info... Very believable except for the small detail missing of "why would Poe want to try and escape in the first place if he'd defected"? Annie's going to need to think quick to come up with a story good enough to get out of this situation! Perhaps mentioning that Poe was tortured by Kylo Ren after being brought up to the ship would be sufficient to assuage the suspicions.
And Pete states the main reason Poe can't be dead; the body was never found! And in the light of an apparent defection, claiming there's no body to find in a supposed death is practically confirming the team switch. This is why you always want to check for the body in narrative settings, it keeps things much simpler in the long run. Admittedly, Jim isn't the best at judging character death, but then again, he was able to pull off the long con of being the traitor in the ranks very easily. So this could go either way I think.
Oh, one other thing Pete could do to show he's obviously out of character? Show that he cares about an NPC beyond their usability.
BB-8: Wait, did you say Poe?
Finn: Yes, it’s Poe’s jacket.
BB-8: Go on...
Finn: The First Order captured him. I was a trooper, but I realised they are not the good guys and decided to defect to the Resistance.
Finn: I helped Poe escape. But our ship crashed and... he didn’t make it.
Rey: Did you see his body?
Finn: No... He was pulled under the sand and there was this huge—
Rey: No body, huh? There’s something odd going on here...
Rey: Corey, you said Poe defected to the First Order.
GM: Are you saying this in front of Finn?
Rey: I said “Corey”. How more obviously out of character can I get?
BB-8: You could show me some respect?