Press L on your keyboard and go to the Dialogue tab. You can follow along with your partner's conversation without all the random chatter cluttering things up.
Here is my Horrorscope's Half-Assed but very Quick Feature solution tip of the day!
Make a new Hotkey that takes us right into this dialog screen.
When a player goes into conversation, on the other Coop players screen, simply state in some manner like in the middle of the screen: "Conversation Started by Partner: Press X to follow along." And have it auto-exit when the dialog is over.
That would be a possible good way to get something going fast, vs a possible fairly sized overall to the dialog system.
Another easy one (Sorry to just whip out the word easy to devs):
In the Dialog Chat the bottom right. Get rid of NPC Chatter or make a new section that has that and Coop Dialog separate. And in that Coop Dialog make sure both the Players and NPC's responses show.