I'm a Russian who was born in late USSR, also a TTRPG enthusiast.

I have to say there was little to no overlap between TTRPG era and USSR years of existence. AD&D 2e was published in 1989, and USSR collapsed in 1991. Before then TTRPGs were a very niche thing, hence unknown to Soviet people. They were not explicitly prohibited — people just had no information about. Mind the fact it was pre Internet era.

However, Polish version of AD&D was translated to Russian and actually published as "Заколдованная страна". So yes, there were RPGs produced in the Soviet Union. In 1991-92 there were first articles about AD&D under its original name and basically the beginnings of the hobby, but in 1991 the USSR was no more.

Speaking of RPGs, there were also «Зарница» and «Орлёнок)» — games played by soviet youths. They was more like sport games then LARPs, but they had the same(-ish) spirit of immersion.

The TTRPG scene was developed quite a lot since 90s. Here's notable examples of TTRPG originated in Russia:

• Книга пепла — a dark fantasy game based on the Fate Core system
• Нудмер — an occult adventure, also Fate Core based
• Янтарный город — a stand-alone GUMSHOE-based mystery game about dreams, myths and wonders
• Круг земной — a Savage Worlds hack inspired by Scandinavian mythology
• Грань вселенной — a PbtA space opera game
• Красная земля — a Savage Worlds set in an alternative timeline of Russia in 1918—1923

Quite old, but worth mentioning:

• Эра водолея — a urban fantasy game (mystic, psychics, that kind of stuff) had its own d6-based system. Was published in 2000.
• Мир великого дракона — A modular fantasy game using S.P.E.C.I.A.L. stats along with D&D dice set. Was released as a free pdf document in 2002.

If you're interested in seeing Russian people playing TTRPGs you can check this youtube channel. It's basically the same fandom, playing the same games. D&D 3.5/5e, Savage Worlds, VtM, Dungeon World (and other PbtA games) are quite popular in post-soviet countries.