By definition, a truly "invisible" watermark cannot be checked. As soon as two copies of the same document differ, the presence of a watermark becomes obvious; if they don't differ, there is no watermark.
As I know there exist technologies that use not invisible watermarks but the steganographic approach and it gives them an opportunity to create millions of copies for each document that will be unique.
> As soon as two copies of the same document differ, the presence of a watermark becomes obvious;
I don't think that part is obvious. Imagine 10 photos or scans of the same page with slightly different cropping. This is quite normal: it's practically impossible to photograph or scan a document and get exactly the same file twice.
This is a naturally occurring watermark rather than an intentional one. Scanned images are unlikely to afford sufficient stealth to detect leaks (the potential leakers know that the image they scanned is unique and others have different scanned images). For the standard setup of distributing watermarked documents to people, differently scanned images are unsubtle and therefore not as useful as, say, doctoring the least significant bits of JPEG coefficients.
Naturally occurring differences are necessary if you want to embed the watermark, otherwise detecting it is trivial - you just need to compare two copies. A photo or a scan of a text document can easily weigh 2 MB, that's enough data to embed a watermark in a way that is reasonably difficult to detect. Embedding watermarks that will survive re-encoding while being hard to detect is far more difficult though.