It's going to be extremely dependent on the school, the teaching quality, and the administration's priorities and flexibility.
Our kids attend an amazing private school that offers no gifted programming and highly discourages grade acceleration. The teaching staff is second to none, the student body is extremely bright/gifted, and the teaching level in general is very high. For example, my K'er last year got second grade math in class with the other bright math kids. He probably could have done a higher level, but he was engaged and happy, and as a social child prefers working with other children. Our school caters to very bright children, and mine have been engaged and stretched in enjoyable ways without any need for pushing.
A very good private will be better than an average public school, and a very good public school will be better than an average private school.
In general, if private school is a financial stretch that will limit extracurricular activities, I'd say it's not worth it and better to find a very good public school. If private school is not a financial burden and you can be assured of teacher quality and engagement, it's worth considering in comparison to your public school choices.
Additionally, an engaging, challenging classroom is always better than a once-a-week pullout. My preferences for kids are: with age-mates getting appropriate challenge, failing that, accelerated to get appropriate challenge, failing that, a pull out. Some children may need all of those, but don't assume a once-a-week pullout will make a difference for a child.
