Idk why but I've just completed 2 missions with 2 separate classes and it still says 0/6 completed on my pros account, does it have to be in a public match?

Make sure every time, that you are logged in into the account while you're playing, because if not, it won't count in your progress.

Mainly, if you play on console. Also, sometimes it can take up to 1 hour for the PROS account to notice your progress.

Just check your account later. Hope it helps. I also learned the hard way.