User @cgrau is the most knowledgable of Soviet era sets, check some of their past posts on the subject. He provides a history of the most notable sets.
I have not seen a designer attributed to any Soviet sets (I'm still trying to find details about Alexey Poni for a different set). Even manufacturer can be difficult to assertain do to pooling of common resources for efficiency (and penal colonies were often used to manufacture sets).
This might be helpful: https://sovietchesssets.com/gallery/
The Chess Empire did a reproduction of the Soviet Shkolnik set below, and on their Etsy page I saw a reviewer mention it’s a 1920s Soviet set. Yet ERWoodLeather.Shop attributes it to the 1970s. Does anyone have more information on when this set emerged? Do you know any prominent chess players who used it? Do you happen to know who designed or produced it?