People living in the countryside could legally own hunting rifles, particularly after the Civil War. It wasn't necessarily easy to get a permit and you basically had to prove that it was a necessary part of your work/livelihood. It was highly regulated, and there were times (right after the Revolution and later under Stalin) when there were crackdowns on gun ownership include laws requiring people to relinquish weapons (or have them confiscated). In 1923 the there was a resolution "On Hunting" that allowed people to officially register for a hunting license that was issued by the People's Commissariat of Agriculture. The NKVD was responsible for the permits for the firearms and ammunition.

I have not seen a ton of research on this. I know that there is a little bit on it in Orlando Figes' A People's Tragedy, and even though I am not a huge fan of Figes, I don't doubt this particular research. The specifics also depend on the period of the Soviet Union we're talking about.

Soviet citizens had the right to hunt and this implies a right to own guns. This right to hunt was symbolic as before the revolution the legal right to hunt had been severely restricted. In 1918 all citizens were compelled to register their revolvers, non smoothbore rifles and automatic rifles immediately and upon registration guns were liable to compulsory purchase. People who lived in the Caucasus mountains were not obliged to register their guns (I don't know why and I don't know whether this actually means that there was no duty to register or a delayed duty).

To gain the right to hunt with a gun you had to join the local hunters union and pass a firearms test. (Thousands of hunters unions were set up and I am not sure how much this was a communal and how much a solitary activity).

If you worked on a collective farm the manager would arrange your license.

The right to hunt implied a right to a gun ; however even if you were allowed to keep your gun it could be confiscated (celebratory gunfire resulted in removal of the right to a gun, for example, as did waving guns around). In 1922 people had to register their guns again (not counting shotguns). In 1924 all trade in guns had to go through the state (except for muskets and shotguns).

In 1940 people had to register muskets (and remember registration doesn't mean that your gun won't be complusorily purchased). In 1953 they removed the permit for smoothbore guns. In 1959 they introduced a permit for shotguns.

Unfortunately I don't know much about archery / hunting in the soviet union.