Indeed... this is how it worked in 2nd edition. You would draw a mission card. You wouldn't know what your opponent had so while you were trying to achieve your objective, he was trying to achieve his. Of course there was the ubiquitous kill points mission so you were still trying to kill each other at the same time.
It made for some awesome types of games where you could figure out what his mission was during the course of the game through his actions (Why is that squad of plague marines just sitting on that backfield objective all day? Why are you shooting literally everything in your army at my psyker?), or you wouldn't know until the end where it was revealed.