Regarding Cuba... well, they do have lower infant mortality than the U.S. so I wouldn't be so quick to mock them.
Regarding "the former Soviet Union"- did you mean, in the Soviet Union they were required, or they are required in the places that comprise the FSU?
In any case, the USSR doctors never vaccinated sick children; they also allowed exemptions for certain cases; and they were slower to approve new vaccines. Presently, they have several vaccines which the US does not have- including BCG, which I am trying to figure out how to get out of without having the local nurse come to my door every day to complain. They have the chicken pox vaccine. Most FSU countries go by the WHO standards but I think they are more wary of needles than we are.
"they could be very rude and pushy!"
Nobody's saying the Soviet doctors were NICE about it. But they do do the forms, they did have the exemptions, they do ask you when your kid was sick and delayed vaccinations for sick children, about reactions in the family, etc. Now, and in the past, as far as I am told.