Agree with above. So much depends on the child and environment that it is really hard to say. Be aware of the signs that things aren't working and be aware of any options you have if you need them. Advocate to improve things even if it seems impossible and/or glacially slow. A lot of it comes down to teachers and relationships. Teachers can make a huge difference regardless of what the official policy is if they are willing and able.
I have two kids with two very different personalities and two very different experiences with school. Each had their own way of letting us know when things needed to be adjusted and we did the best we could. What I've learned from both of them is to take it year by year. They change, their needs change and you just do the best you can with the available options.
Expanding on looking at the bigger picture I'd also consider what might be lost by moving or changing to private especially since both are pretty big steps. Maybe it means a longer commute for a parent (or two) which cuts into family time. Maybe it means less money for extra curricular activities or travel or whatever. What can you do outside of school to challenge her if school isn't? Is she interested in sports or music or something else where some of those skills might be developed?
I'll also note that gifted programs aren't always the perfect solution for all kids. If your child happens to be a good fit for that particular program then great but there are huge varieties of gifted kids and gifted programs out there which means that there can be many kids who aren't a good fit for that particular one. The links provided above are a great start.
