I won't get into the liability thing, but yes, a loose screw in some drain designs will allow water to leak. On some, that screw is hidden, so I don't see how you could be liable...you would have no way of knowing it was leaking unless you were downstairs. If they waited a long time to report it, THEY may be liable for damages, since they knew about it.
It is the landlords responsibilty to provide a working set of facilities. Unless you loosened the screw and did some maintenance you weren't authorized, I think he's on the hook...I'd fight it. (but, then again, I'm not a lawyer nor do I have to live with the landlord).