I have had many friends who are from “over there.” In regard to what Igor said, the two most important characteristics are:
A: In Soviet Russia, nationality went by the father, and up until emigration became possible it was much more advantageous to be a Russian, so what was listed on the identity card wasn’t necessarily true by our standards.
B. Once emigration was allowed it became better to be a Jew, because if one member of a family was Jewish, the whole family could emigrate. I heard many funny anecdotes about this situation.
The bottom line is that the situation is often very confused. I knew one poor man who after extended arguments and persuasions got his son to agree to a bris and a bar mitzvah, and then found out that because his wife’s mother had been a Russian married to a Jew, that the boy wasn’t Jewish. He was heartbroken, because even though he wasn’t Shomer Shabbos, he took his Jewish heritage very seirously.