The rights/permissions are probably just fine. Most likely your local server is running on a nonroutable IP address, such as 192.168.x.y or 10.x.y.z. Serendipity then makes links using that address. When a local machine tries to connect, it has no problem, because it's on the same network. When a remote machine tries to connect, it looks for 192.168.x.y in its own network, not in yours. Naturally, it can't find the stylesheet.
In you configuration, if you change the "Auto-detect HTTP host" setting to "true", Serendipity will do its best to use the same URL your visitor used to access it. This should solve the problem.
