The mother of four gifted students, all went through the public school system. Some years were great, others were not so great. There's pros and cons to whatever you do. We live in a smaller community so it was important that they had the other opportunities as well, such as social, sports, arts, that were primarily only available through the school system. Was it the best choice? We'll never know, but three graduated as valedictorians or top five, two scored the highest of their class on the acts, one holds the school record. So I guess the public school system didn't completely fail them, although some teachers along the way did.
Here's a few examples: Math grade 6, ds is above grade level, teacher gave him the book and basically said have at it. Science, grade 6, ds is above grade level. Well, he's very bright. dd grade 6 is tested as geniouse, we could put her in an enrichment program. dd grade 9, math teacher is incorrect and argues with student to the point of frustration, dumb. dd grade 10, english teacher doesn't know what to do with her, allows her to develop her own curriculum, ok fine. But it went on like this forever, and there was only a time or two you would find a teacher that truly appreciated your childs intelligence, for the most part they are intimidated by it. Forgive me if their mother doesn't spell like a geniouse!
Have one to go, this dd doesn't have the social stigma that can go with being gifted, she has all the common sense in the world to blend in, yet retain a 4.0. Lucky for her, most don't have that which makes the public school a not so fun place to be. While the other three were the ones the teachers wanted to take home and keep, this one is the one that the teacher would prefer to send to detention. Oh my. Isn't it all fun. Yet, her attitude allows her to survive.
Funniest thing, the only B dd #1 received was in penmanship in grade three. How funny is that? She still writes like she's in third grade, she just writes beautiful words now, as she has a BA in english.
Good luck with your decision, I know a lot depends on the resources you have available to you within your community.