I don't see what the one has to do with the other. But I need to be more consistent with trying offers for “Declare war on...” to AIs to see what they would pay me -- because I did not think that was really an option!
I have used the negotiation screens for mutual DoW, but when I click on the “what will it take button” it has always resulted in me throwing something in. Otherwise, I have been mutually DoWing from the dialog screen before that, because I or they want 10 turns.
If, on their turn, an AI asks you to DoW, does that mean they will almost certainly offer you something to DoW through the negotiation screen (if you wait until your turn, and negotiate a Dow)? If I actually like the AI-initiated DoW offer (which is unusual in my experience), I have been hesitant to pass it up.