At least around here, most private school students are not from the community the private school is in- they essentially commute to school. In that case, they don't pay any taxes to our city. The public and private school swim team uses the community center pool, but I don't know what either pays for it.
However, the kids on the private school swim team can't use the pool outside of swim team time unless they pay the (large) non-resident membership (or happen to live in my city, but that isn't very many). The kids who go to the public school get free use of the pool outside of swim team time because it is tax supported, and they are residents.