I was an Altar Boy (in Latin Mass days). I did attend Catholic School but the Nuns had nothing to do with our instructions. We were trained by priests only. The classes were on Saturday morning and although most were students at the school there were a couple who attended the local public school.
This was certainly different from my experience. As I stated early, in my Diocese the overwhelming number of alter boys were trained by the Nuns, but had to be reviewed by a priest before being awarded Alter Boy status.
I cannot speak to your experience, only my own. I never remember any Public School kids, no less Private Secular kids becoming Alter Boys