Any child that has gone to preschool will most likely have been exposed to what kindergarten offers. Our parents generation didn't have Mother's Day Out or preschool as readily as we do, so our kids are exposed to a lot more at a younger age. Now that most districts offer full-day kindergarten as opposed to half-day, they're finding out that in 1st grade they are more advanced and flying through the curriculum months ahead of what 1st graders learned 5 years ago. There are many children that attend kindergarten with no prior schooling, so the curriculum has to be one that is age-appropriate, not child-specific.
My son attended K already knowing letters, etc., but as one other post stated, there is much more to kindergarten than just ABC's and shapes and colors. It's a transition year, and those wonderful teachers nurture the kids and get them ready for 1st grade. They teach them class rules, how to order their lunch in the cafeteria, how to go to the school nurse, how to tie their shoes, how to use a locker, how to use the library, how to find their classroom by themselves - really it's about learning independence, not academic skills. They get involved in school-wide functions such as parades, Spirit Day, pep rallies, Field Day, etc. which encourage school pride, a value I don't think is present in private K. Kindergarten is about having FUN and learning to love school, so that when they enter 1st grade they're ready for "school".
They learn to spell and read in kindergarten, so if your child is ahead of the curve, he/she will be place in a reading group specific to her capabilities, not the rest of the class. There is plenty of opportunity for the teacher to assign more challenging work to your child if necessary. Most people I know put their children in private K if they need extra time and attention, not the other way around. I say, if they're ready, send them to Kindergarten at the ISD!