Short answer: copyrights got transferred to their respective successors.

Long answer: In general, the Soviet Union's attitude to copyright, was, in a word, lax, in part due to the fact that so much was owned by the state anyway.

There were also plenty of exemptions to copyright. For example, even translating a work into another language meant that the author of the original work didn't hold the copyright. The government reserved the right to forcibly nationalise any work, and even broadcasting someone's play in a theatre or concert was ruled fair use.

To add to all this, despite the Soviet Union's founder Lenin himself starting his professional life as a lawyer, the Soviet Union by and large lacked a litigious culture. Soviet citizens and organisation in general preferred to avoid courts.

Now, with that in mind: as part of the general post-Stalin shift towards the international community, the Soviet government decided to follow international copyright conventions from the 1970s, with its signing of the and implementation of the Universal Copyright Convention of 1973.

Before then, as mentioned above, the USSR had only its own internal copyright law, and agreements with a couple of other socialist satellite states. And as so many things were the property of the state anyway, in practice, the infringing party and the copyright owner were often the same party - the Soviet government, as u/MaesterBarth notes in this separate thread.

Then, on the dissolution of the Soviet Union, state property very often became private property - and technically, films made during the Soviet period became the private property of the now mostly privatised film studios such as Mosfilm and Lenfilm.

This meant that film studios were able to generate royalties from their Soviet-era content. A nice little earner - if you wished to bother chasing down royalties in the freewheeling 1990s.