If parents perceive the public school is not providing suitable services, and the private school is, then the public school finds reason to correct the deficiency - competition works.
And if the private school can't handle the inundation, then there's incentive to either expand capacity and/or another school adapt/open to share the load.
And if the private school can't handle the inundation, then there's incentive to either expand capacity and/or another school adapt/open to share the load.