They shouldn't have to go to private schools or sit at home and learn by themselves. The government is supposed to supply a good education system, and it is failing for some people.
In some places, there is something called a Gifted and Talented program, or Academic Challenge, or whatever they decide to name it, where students are actually encouraged to learn independently and think.
Other places satisfy government requirements for the program by allowing students to take a class that prepares them for the Academic Decathlon competition.
The fact is that public schools are capable of getting the interest of the advanced students, it's just that they don't try.