If you look at workshops in the civlopedia it's listed as giving +2 production and has a citizen slot. These are two separate things, the +2 production (and the gpp incidentally) is a passive yield that doesn't need to be worked. If you do work the citizen slot he earns +2 production. All the industrial specialists give +2 production, all the encampment specialists give +1 production +1 culture and so on.
Edit: The adjacency bonus is a passive yield too by the way.
No, Specialists only generate the Flat Yields that are displayed in the UI, they don't increase a districts output, they are not required to get the output of the district, and they don't generate Points towards Great People.
Overall, Specialists are pretty terrible, and more an "Okay, this city is done growing."-option than anything else.
To be fair though, make the yields too high and they'll become an "Use always!"-option like with Scientists in Civ V, that was also terribly boring design.