This has happened in a couple of missions and I've bugged it but I want to put it here as well.

I wish I remembered the name of the mission where it was awful but it was the one where you have to escort Ambassador Sakkoth.

So you get into the system and you get a popup the klingons are saying blah blah he's a shapechanger. So you get out of the dialog and try to engage the klingons and you get the popup communication again. Get out of it, head for the klingons, BAM get it again. It happened at least 4 times.

Then I engaged the klingons and it kept happening, it was even worse in combat since the incoming message immediately drops my speed to 0 and takes up a huge portion of what I'm looking at, I got this message during combat at least 5 times and probably more.

Then as I destroy the last klingon there is a loot drop for me. I get the popup saying beam down to the planet but I don't want to do that because I want my loot. So I get out of that dialog and then try to grab my loot which I can't. Can't get button to take items, ramming right into the loot changes nothing, flying away from it then back still can't pick it up. So I give up on the loot and now I try to beam down again, no go, no dialog boxes, flying to the planet doesn't help. So I have to return to sector space then fly back into the system to get my beam down box back but I never did get my loot.

This popup problem happened on a couple of missions that I saw, with the multiple dialog for the same thing and the not being able to collect loot after a dialoug box appeared. It was very frustrating.

I had that same problem. I suspect any time a member of the instanced team joins and views the quest dialog it appears for all members.