I actually tested this. Loading a quicksave with a craft in orbit in a DMP server doesn't put the craft on the server - other players can't see it, and rejoining (and thus re-syncing the crafts with the server) causes it to disappear. The server syncs science (and all other stats) in the same way, but doesn't verify if an action taken by the player could actually have been legitimate.
The server knows when I take measurements, when I recover vessels and how much science I get from that. It should keep count as well, and in that way it would be easily possible to detect if the player has tampered with their stats. Every properly made multiplayer game verifies client messages on the server to make sure clients aren't cheating.