I found an imperfect solution: apparently applications that use their own full-screen mode (an older API maybe?) and not the OSX full-screen mode, disable the Notification Center while they're in full-screen. Scrivener's composition mode does this, as does Word 2011.
A solution that worked for me in Scrivener was to go into regular full-screen mode (View->Enter Full Screen), which uses the OSX full-screen API, creating its own Space for Scrivener, putting the blue full-screen icon on the right-hand side of the menubar, and leaving Notification Center intact.
If I then go into composition mode (View->Enter Composition Mode), all is well and Notification Center works.
Note, this doesn't work in Word, because in its infinite wisdom, it kicks itself out of OSX fullscreen mode, ditching its Space, before it enters composition mode. So I still don't know how to receive notifications while composing in Word, but maybe that's just another reason not to use Word.
Thanks to njboot for the screenshot showing it was possible to get notifications in Scrivener composition mode!