Interesting because my son's private school K teacher said in 30 years, she'd never had a student enter K already reading and accused him of being a liar, in front of the whole class. The beginning of 100 reasons why we left private school.
It really truly depends entirely on the personalities involved and the school itself. I think for every story one way, you will find one the other.
For example, I read through and saw Luconu (I think) was going private to get a grade acceleration. Our private school wouldn't even remotely consider it. Public school did it in two months.
We found that the private school had an ego problem. They touted themselves as a college prep school for advance children and behaved as if every child should be lucky to attend. They wouldn't discuss anything that made them evaluate their program or even how they worked with our son. They, in their own review, were fabulous as they were.
The public school hasn't been easy but they've at least been willing to discuss. The teachers, minus one this year, realize they are undereducated about gifted kids and are willing to learn and eager to try harder for DS.