Have you looked into the backgrounds of people who currently occupy the positions that you want? Did any of them start out in a private school? Do you have any common acquaintances who can make an introduction so you can get their sense of the value of the position you were looking at?
What happened to the guy before you? If he left for another job where did he go? Any common acquaintances with him?
If you decide that this position isn’t worth a pay cut, or at least as big a pay cut, don’t turn it down, counter offer. Express your interest and enthusiasm in the opportunity but explain that the numbers don’t work for you. If they like you, ther could be movement.