You should hand out their faction mission handout at the beginning of the scenario. The faction missions generally do a good job spelling out what a player needs to do to accomplish a faction mission. If you're playing with a regular group who is new to Pathfinder Society, you might give them hints in the first scenario to get them into the spirit of it, but after that rely on them to accomplish their missions.
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Absolutely ... once your players are used to it, the responsibility lies completely in their hands.
As a veteran player of groups that are fairly consistent, I would also like to point out that sometimes the DM needs a reminder. While we would all like to believe that we have perfect memories, and that our DM always knows what's going on, sometimes they will forget the precise details of five different faction goals, of which there are occasionally two per faction.
So ... sometimes, it's even a good idea for the PCs to give the DM a reminder: "BTW, when [my PC] does [such-and-such], he takes time to do it [in this specific way]." Making a point of the action is usually enough to jog the DM's memory so that he takes note of you completing your faction goals. So it's fair to mention to your players to do that if needed.