A view from the 'other side of the fence', from someone who could afford to use private schools but haven't chosen too (so my DCs have many of the 'other' trappings of privilege in terms of house, ability to holiday, pay for extracurricular activities etc ... not cars, we run cars for green-ness and economy, not image)
We live very close to a large, floridly Victorian Gothic private school with prep, and routes to my DC;' primary and to other daily activities led past its extensive playing fields, astroturf, sports hall with swimming pool etc etc. So my DCs, from their earliest years, could see that there was a school very close to their very normal state primary that was different and had different facilities (the private school is, incidentally, where my DH was sent to board from the age of 7, something he is profoundly affected by). So very very early, in general conversation, we explained that some schools are state funded and free at the point of delivery, and some schools are paid for directly. Also, as they grew older, we talked about what that money clearly bought - great facilities, sports fields etc - and about what it might not buy - particular excellent teachers from their own school, friendships, character traits like kindness. Still older, we talked about why we choose to send them to state schools - better results, better provision in the community rather than in the private schools for their particular 'things' - and how lucky they anyway are in comparison with other children whose state schools may not be as good, who may not be able to access or pay for extra curricular activities, who may not live in a nice house etc etc. I have taught in very deprived schools, so they are aware that even within a few miles of where we live there are children who are hungry and neglected.
They both mix with privately-educated children at a variety of activities - sport, dance, music - and don't regard them as 'privileged', just as children whose parents have made different choices. They would regard themselves as lucky.