In an article Zhukov wrote, he mentioned a rather off hand that Bulganin, trying to curry favor with Beria, disbanded the Red Army soccer team TsSKA, and sent all the best players to the Dynamo Moscow, which was affiliated with the NKVD. It was a casual aside, part of his description of internal corruption on the part of Beria and Bulganin, who he hated, but got me thinking...
If these sports teams were affiliated with government agencies, what happened to them as they transitioned from state run organizations to private ones? Dynamo is now owned by VTB Bank apparently, so at some point it got privatized, but Wikipedia kind of skips from the '70s to 2008. Soccer teams were what sparked my interest here, but I imagine similar answers would fit for Hockey clubs, so I'm cool with that too!
CSKA Moskva (the umbrella organization-soccer, basketball, handball, etc), known as the Red Army team in the Soviet Era, still exists as the separate teams, and are among the top teams in Russia in their various sports (at least in soccer and basketball). They were under the control of the Ministry of Defense. These teams have transitioned to private ownership and board of directors, as is similar for many European clubs. That said, as of 2009, ownership of the CSKA soccer side was split between Blue Castle Enterprises (49%), AVO Capital (26%), and the Russian Ministry of Defense (25%). So the Red Army retains a connection to the historic Red Army team.
edit: I couldn't find historical ownership changes, but it remained a Red Army side until the dissolution of the USSR, and I imagine that the Ministry of Defense has held a relatively stable amount of shares, and the 25% is probably a result of a sale in the immediate aftermath of the dissolution, but that is conjecture
This is something I don't think has actually been written about, at least in regards to hockey, so it is more of an educated guess. But most teams in Russia are still owned by governments, large corporations or something like that. Details regarding Russian hockey team ownership is notoriously difficult to find accurate, detailed information on, so its hard to know exactly what's what there. But I'd expect that much like other Russian business interests, teams that were sold were done for cheap to various oligarchs, though like I said I can't confirm that, especially as most KHL (Russian hockey league) teams are still owned by local governments, or subsidiaries of the Russian government (both Gazprom and Rosneft, for example own teams).
However the one example I can provide information on is HC CSKA Moscow, as they were a rather unique example. Control of the team, but not necessarily ownership, was sold to Nugzar Natchkhebia, who embezzled most of the transfer money NHL teams paid for CSKA players.
With the team broke and player poorly, a 50% interest was sold in 1993 to Howard Baldwin, at the time owner of the NHL's Pittsburgh Penguins (and previously the owner of the Hartford Whalers and Minnesota North Stars). The team tried to import some Western arena concepts, like ads on the boards and sponsorship, but the whole thing was a financial disaster for Baldwin, who sold off his stake within 2 years.
PBS conducted an interview with Baldwin about the whole project some time ago that you can read here. Its quite interesting, even if it doesn't fully answer the question. Sports Illustrated also has a few articles relating to this; they mostly talk about the corruption and troubles hockey players faced in the early post-Soviet years, but it provides some insight into how everything was: Russian Revolution Thanks to capitalism, sports in the former Soviet Union have changed -- for both good and bad and Running on Empty With its best players now in the NHL, the former Soviet Union's hockey machine is out of gas.