I'm pretty much OK with other parents disciplining my children in public. I quite like it, as long as it is fair and firm, not rude or out of proportion. For example, at our local playground my DS can sometimes be a bit boisterous and occasionally all he needs is a 'hey there, young man, you watch out for the younger children/don't push/wait your turn' etc from a well meaning mum or dad (if I havent spotted his bad behaviour).
I even court it. Today in the supermarket DS was being really over excitable and silly and kept bumping into people because he wasn't looking where he was going. I told him to look out twice, and eventually saw an old man looking very grumpily in DS's direction and said 'Look, you see, DS! That gentleman is going to tell you off in a minute if you don't behave'. Worked a treat.
It takes a village to raise a child and all that...
I don't like sighing and tutting and sarky remarks aimed at young children, though. A few times, when DS was younger, he had some truly terrible supermarket tantrums and we got some really unhelpful (and even downright rude) 'input' from grumpy passersby which left my blood boiling.
It is important to remember that we all exist in the same world. Children have the right to have fun and to be kids. We are not living in the Victorian age, and a spot of dancing and twirling is not a crime. However, there are busy working people, fragile older people etc out there, too, and they also have the right to shop in peace without being floored by a junior ballerina .