What is your take on outsiders telling off your child in public? Do they really have this kind of duty towards our own children?

It really depends on how your child has behaved. When I see a child being harmful to others ( not only to my kid) and the guardian is not there to reprimand or restrain I tell the child off in a firm but gentle way. When we are out, I try to keep my eyes on my child all the time to make sure he does not hurt others or be a nuisance to anyone.