My 3 boys are in public schools. The school system is a good suburban district with about 18,000 students.
2 of them have been identified as "Gifted" through 3rd grade testing, and were invited to a self contained gifted class at a school different than their original elementary. The testing occurs in 3rd grade because that is when the data shows the IQ testing is a reliable predictor. When we received the first invitation we encouraged my son to accept, because he was very very bored at school. It had become a grind. He also did not relate well with kids his age. He related well with adults, but struggled with his peers. We hoped he would meet other kids like him and make friends. At the new school, their class size shrunk and the resources grew. Class went from 26 to 16. He made new friends and was able to relate with the gifted kids. The new school was a huge success and fit him like a glove, so we encouraged my second son to accept as well. My second son would have been fine in the regular elementary, but because we really like the gifted school we encouraged him to go. My 3rd son has not taken the test (currently in first grade), but he also is flourishing in the regular elementary school, both socially and academically, and we have no concerns.
By way of example on resources, my 5th grader gets bused to the middle school to take Advanced 8th grade math, then bused back to the gifted class. This happens 5 days a week, and he is the only kid on the bus. They have science fairs, invention conventions, chess clubs, lego robotics, etc...
All this to say, our public school system has significantly more resources than our local private schools. The gifted school teachers are all specially trained and do very well with this group of kids.
I can't imagine a better scenario for us, than what we have through the public schools. I really do think the issue of public vs. private vs. home school is best addressed on a case by case basis.