I don't know about Russia, whatever the color, but in White Ruthenia you need to show your passport (and probably visa, too, which is a sticker in the passport) to buy a prepaid SIM.
Funnily enough, although I used to go to Belarus quite often, it's about the only place where I haven't bought a SIM, in part because Velcom used to have extremely cheap roaming (back in the days before Vodafone "standardized" roaming costs) and prepaid was rather late in arriving there (and when it did, it was initially expensive and a pain to use, and you had to buy a phone - much like may other places at the beginning). But friends from there tell me it's not a big deal to get a SIM these days.
http://wikitravel.org/en/Belarus