The info is determined by perception. Anyone can do it, though.
ForkTong
One thing I've noticed in the previews that really threw me was that when one of the players that isn't the client (i.e the other guy) says something and it's the end of the conversation, you don't get to see what they said. I know this is all still stub state, but in general I just want to say it would be nice to have a chatlog, both for things like this, to refresh yourself on a conversation that just happened, and to keep track of all the floating strings that NPCs are saying (especially since this is a game with no voice-overs).
I think they said that there will be a conversation log so that no one will miss anything, even if the conversation was between an NPC and the other party member across the map.
But yeah, there's a couple quirks with the conversation system at the moment, mostly because the second person to participate in the conversation is the only one who can really see what everyone's thoughts are. As you noticed, that was clearly a spot where the first player should see what the second one said and then have the [End] option.
The other change I hope gets implemented,
ForkTong is to let people in a Decision dialogue to be able to change their mind after hearing what their partner has to say.
Right now, only the second person to talk (or the AI) actually has a choice about whether to agree or disagree with the first person. The first person makes a decision and is stuck with that decision – before even knowing what the second person’s opinion is, but the second person can see the first person’s decision before deciding whether to agree.
No one can ever actually be swayed by the other's arguments and change their mind. This one is easy to change, with just a generic "Fine, we'll do it your way", alongside the three "No, we'll do it my way: [Charm] / [Reason] / [Intimidate]" options.
(To be clear, I think this should only affect Decision dialogues, not Opinion dialogues. There's no need to do anything with the single-stage "was he smart to jump from a cliff" / "Is that girl sleeping around a slut" / "Was that ghost lightkeeper a jerk" opinion dialogues.)