So I was playing last night with my brother and we completed the mission but it would not end. I've seen it mentioned in very old threads about having to call everything in and get all the guns but at one point we had a 3rd person on the server with us and when the mission would be complete we would go on to the next one with stuff missing.

I tried pressing every damn button I could think of but nothing. Saw nothing in the menus as well. This was on a co-op server found in the browser list within game.

Indeed, the session will automatically end when all objectives completed, all reports to HQ and evidence collected (on certain maps it can be drugs, not only weapons from suspects).
Either way you can end mission when you go back to start point or other exit (the same ones what used for entrance on planning phase).