Propaganda Center quests are the only side missions that are not unlocked via liberating outposts and Bell Towers. In fact, in order to unlock these, you will need to destroy the Propaganda Posts. Once you’ve managed to destroy five of those, you’ll unlock the first of four of these quests, with more being unlocked after destroying 15, 25, and 35 posters.
The entire goal of these side missions is to destroy the propaganda equipment being used by Pagan and his Royal Army. You will go to these centers, where there will be a few enemies, and destroy the equipment, which will flash. This will include computers, boxes of posters, printing presses and even some vehicles. Most of the items will be in buildings, which will be marked on your map with a yellow shape.
As you destroy this equipment, two things will happen, one of which will be the yellow bar at the top of the screen filling up. Once this is full, you will have destroyed all equipment. The other thing that happens is that reinforcements will show up to put a stop to your madness. They will come about two or three times and may include enemies like Heavies or even a helicopter, so be prepared. The missions will end when you’ve filled the bar and eliminated all enemies.