I've no idea whether there is a great divide re state and private parents and sporting events because I don't think they meet all that often.
I live in a rural area where the vast majority (even the very rich) send their DC to the local primary and then the comprehensive. It is a proper comp with no grammar in the county or within reasonable travelling distance.
There is a very selective private 4-18 school in a market town, and a much less selective/sporty private both of which occasionally compete with local state schools.
The worst behaviour I have seen was at an inter schools secondary cross country was when parents of the state school pupils actually booed the private school DC. Cheering your own son and his school friends on is one thing but jeering/swearing and yelling don't let the XXXXX school get past was awful to listen to. I don't know if any of the private school parents were there to here it.
One of my sons did play football at the sporty private in year 7 and beat them 11-0. State school football team were never invited back and private school parents were not pleased. They did give us all tea, coffee, squash and biscuits in their very nice dining hall though.
There was a regular swimming gala between the schools too. The state school always won the girls competition and the private always won the boys competition. The sate school could never host it because they don't have a pool. We got free coffee and biscuits again.
DD played the private school at netball and tennis in the primary years. There were lots of state primaries taking part and the private school were the host school. DD school came last in the netball but there were only 7 girls in her year so team selection was easy! They did better in the tennis.
Normally state school compete against other state schools.