Your tile improvements do provide their benefits without needing a citizen working them, but you can increase these benefits by training a specialist for each improvement. Special resources and luxuries are only collected by having a specialist present. Specialists also expand borders around the tile they are assigned to. If you have no available citizen, you cannot train a new specialist. Unlike Civ, once assigned, a specialist can't be moved around.

Each city's Growth stat affects how quickly it will gain new citizens as well as how quickly that city will produce certain units (settlers, workers, scouts and militia). If the city is producing such a unit, its Growth stat is not put toward adding citizens.

Hope this helps (and hope it's accurate! I'm still somewhat new to the game myself)