How can I separate a bilingual PDF document into two separate PDF files?

There's no easy way of doing it. One possible way is by using a script, like this one I've developed (Custom-made Adobe Scripts: Acrobat -- Split Pages in 2 Parts and Combine to a New File), but the results might look quite odd because each page is going to be very narrow. You would also need to manually adjust the bookmarks in the file, but that shouldn't be too difficult.