I don't know what sort of private schools you and Matthew are speaking of. I strongly suspect Ekim is speaking of protestant schools as he hasn't clarified for me yet. Schools are either going to be devoid of religion which isn't acceptable for Catholics, or they're going to engage in some form of religion practicing. For Catholics, any sort of blended non-denominational crap is unacceptable. Thus, I don't see how any private school that isn't Catholic can be acceptable to Catholic parents.
I'm not speaking for Ekim's example here as he said it was a public school and hasn't mentioned private school AFAIK. (The confusion being that one doesn't normally have the liberty to "shop" through 15 different public schools and would almost certainly end up with children in 3 different schools, not one.)
However, I don't believe a non-denominational private school is nearly as bad as the other options. There's a family here who I hope to become increasingly better friends with. The children will be enrolled this year in a classical academy. They so far seem to me to have an above-average understanding of the Faith which I presume will continue to be taught at home. While I'd much prefer to see another homeschooling family in the area, I'm certainly not going to shun them for choosing this school and hope that it works out for them. The school is new, so its reputation is a blank slate except that it's a spin-off from another state. Time will tell what sort of students populate this school, but even in the best of situations, our hypothetical friendship will forever be limited b/c (1) we don't live very close to each other and (2) our flexible time is mostly during the day while they are in school.