It is hard to give advice without knowing what the game's like, but, in general:
- In cooperative games, clients are usually allowed to check their own collisions, notifying the server when they get hit (and by what), as you want the game to favor the players and for players just generally have fun.
- In competitive games, all checks and additional logic are usually done on server, as you want the game to be as fair as possible and to leave fewer opportunities for anyone to cheat. Exceptions are sometimes made for instantly-hitting projectiles (laser beams, etc.), where damage is dealt if the player was certainly aiming right at the enemy on their side.