The very elaborate, meticulously-edited AMVs set to popular music that you see from creators like Nostromo are, as far as I know, mostly the domain of non-Japanese fans. Naturally, I can't prove that there aren't any made by Japanese people, but I've spent my fair share of time bumbling around on Niconico, and haven't seen anything of the sort.
What Japanese fans do create are what they call "MADs" (for more, see this question). Though (many) MADs are derivative fanworks based on anime, they are, for the most part, not at all like the AMVs produced by Western fans. If I had to characterize Japanese MADs, I would point out the following features that distinguish them from non-Japanese AMVs:
- The audio tracks of MADs are frequently made by editing the audio of the show from which the clips are drawn
- Existing musical pieces (pop songs, whatever) are not used in MADs as frequently as in AMVs
- Whereas many AMVs use footage from multiple series, MADs are more likely to use content from just a single series (or, perhaps, audio from one anime and video from another)
- AMVs are often strongly identified with their creators (cf. credits footage, etc.) whereas MADs are relatively more "anonymous" in the sense that creators are less wont to plaster their name onto their creations
- MADs are typically solo productions (or, at least, the work of few people), whereas MEP AMVs are fairly common.
To get a better flavor for the sorts of things that are typical of MADs, one might like to take a look at the top videos tagged with "mad" on Niconico.
In sum: yes, Japanese fans create audiovisual productions using footage from anime, but no, these productions are not very similar to the AMVs produced by non-Japanese fans.
Many anime conventions in the West incorporate AMV competitions; I suspect that this is one of the major drivers of AMV production in the West. I am not aware of any equivalent competitions in Japan. Also, for English speakers, there's at least one thriving online community for AMV creators: the Org. Again, I am unaware of a Japanese equivalent, beyond general-purpose video sites like Niconico.
(Note: this entire answer is only about the state of AMVs/MADs nowadays - I have no idea what it was like in the past, pre-Youtube/Niconico.)
I had never heard of MMVs prior to reading your question; having looked at a few on Youtube, I think (but am not sure) that these are also uniquely Western. (They also mostly suck, for obvious reasons.)