I do not have direct references, but I talked to Polish people on the net, and they can tell you that indeed, strangely as it may seem, they had private schools back then during the Soviet Union.
This might help as it refers to a kidnapping at the St Augustine School in communist poland in the 50s.
http://www.brightreview.co.uk/ARTICLE-A-Kidnapping-In-Poland.html
Possible, yes, but Poland was not the Soviet Union, to which you referred in the initial post.