They absolutely do. The fact that you CAN see this "aim down sights" animation (and it looks good from a 3rd person perspective) is THE reason they have to have two models. Again, because "aiming down sights" in first person requires unnatural twisting of hands to make it look good for a first-person-view camera, so the same model can't be used.But you can see the other players aim down sights on your screen. That's why I'm thinking they don't have a separate model...they don't need one because they still have to animate the "aim down" right for others to see.
Actually, scratch that - perhaps I misspoke - the same "model" could be used, but the rigging can't (so you're still effectively rendering two different things). However generally, for most multiplayer FPS games at least, the 3rd person model WILL be different from the 1st person one, because the level of detail differs; in 1st person you need to render hands and a weapon (if the game in question doesn't "show your own body" from this perspective), so that gets more attention to detail (plus detailed reload animations and such).