The spotter groups are arranged by Rusadventure: not just for DME
but also for SVO
and a few other smaller airports in the area. All those tours can be cancelled last minute, so be prepared for last minute changes to the plan.
Having been there twice, 2010 & 2012, both with Aeroprints.com (sales agent for Rusadventure). I can say spotting there is no real problem. Just be sensible.
you can watch planes from the large windows in the terminal, this also provides a nice (distant) view of the storage dump. Me and another spotter kept a low profile and had no problem. Another group of spotters went all-in flashing their cameras and scanners: they were taken in by police but a few hours later they were out again without problems.
Another option at DME
is the parking area to the left of the terminal (when facing the terminal). The planes fly over low enough when taking of from 32R. I would guess it is also good for landings on 14L, but I did not get to see that so it is possible the fence will be in the way.
you can observe the planes from one of the terminal restaurants. The restaurant was closed when I was there, but it was late in the evening so that was expected. The pictures from the rest of the group were nice though.
is very good as well. You can easily walk from the terminal into the final approach of runway 24. You are in the middle of nowhere and when I was there, in a group of 4, we only saw 3 other people during the whole day. All of them airport employees. For half the day there was a police roadblock just a hundred metres away from us. The police must have seen us, but they did not care. So it seems this place is legit.
The museum at Monino also HAS to be on your to-do list! Getting there can be a bit tricky, but it is well worth it.