As someone who went to Chinese schools, the hierarchy (at least in my city) was:
Kindergarten, Elementary, and Middle school: private schools were better, or perceived as better. People actively vet their schools if they can, my parents went to 3-4 schools before deciding to send me to one. There used to be a test to get into Middle schools (7-9th grade), but they abolished it, so now parents try to "donate" their way in if they can or somehow pass the interview process. Some public schools were good, but some perception was you could not control for quality, so if you can pay a tuition then you should find a school and pay it.
For high schools (10-12th grade), the public schools reigned king. This is (I think) because you have to take a test to get into. Some private schools are good, but the very very best (multiple kids get into Peiking/TsingHua/Ivy League etc) highs schools are all public and purely meritocratic in terms of getting in.
There are international schools, but those are only open to foreign nationals.