Depends on how that game is made and what the mod change.
I think I can arrange probably all mods on 2 distinct groups: UI mods and Mechanic mods.
UI Mods change one the appearence of the game. Like textures, chat mods and etc.
Mechanic Mods and or change some behavior of the game. Like adding new drops and blocks, add a new monster, add quests or economy to the game.
Most of (if not all) the UI Mods change the game only for you. Other players don't know or see what or how you see the game. This kind of mod don't need to be on server side, only on client.
For Mechanic Mods it a bit different. On singleplayer all the mods need to be installed on client, of course. On multiplayer, some changes on the behavior of the game don't need additions in the client, like altering the rate of droping some item, or changing a monster spawn time. But some other mods need changes in client. If you need to add a new monster you need to add it to the server. But need to add a new model of this monster in the client too.
Like Kikaimaru and I said, its depend on what is the game and how it was made.