I Have a 1 wheel drive 2003 Ural tourist. I drive it in the snow every change i get. I live in the Applacian mountains and the one wheel drive version has no trouble on twisty mountain roads covered with 3 inches of snow, as long as you have a good knobby pusher tire.
I have done many stream corssings, and driven on some muddy rutty roads without getting stuck. I have turned around to avoid huge swampy sink holes though, so some common sense will get you a long way.
With the exception of sand or really deep mud, you do not need 2 wheel drive.
Like has been previously said, there is no differential, so the sidecar wheel MUST slip when 2wd is engaged. The drive ratio of the sidecar wheel is LOWER than that of the rear wheel of the bike, so if the sidecar wheel can't slip you will either drive in a circle or blow out your tranny. I know Ural owners that had the latter happen...
If youre going to go mud bogging or ride through the desert, get 2wd. If not, save some money and get 1wd.
I have 'dual sport' style DOT tires on my Ural, but I have a very aggressive full knobby on my spare wheel. That way i can change to my serious off road tire if I plan on getting really muddy. BUT, you have to plan ahead and switch tires BEFORE you get stuck...