First person shooters can be (and I believe usually are) done server side. Official movement, firing, health, and overall physics is handled 100% on the server. To smooth things out though, the client interpolates its view of the world.
For example lets say you fire your gun at a person right in front of you. The server would have final say of whether you actually hit the target. Rather than waiting for a round trip to the server, your client may choose to immediately show some blood/shrapnel effects to give you the illusion that you hit them. If you actually did, the server will handle the health deduction. If not, you probably wouldn't even notice.