It updates instantly, you don't have to wait for the conversation to end. You see it in real-time.
Right now the only two things this doesn't work for are:
1. You don't know your partner started conversation unless they tell you.
2. You are pulled out of the journal to the convo window if a co-op decision is needed.
Other than that, it works just fine for both people to see the entire conversation as it happens.
