Can you play sports at a public school if you go to a private school?

It depends on the school and the public school district in your area. My understanding is that a co-op has to be formed. This is an official understanding between the administration of the two entities. For example, in our district, the public schools will only agree to co-op in sports in which they need players (e.g., Wrestling and track) or that are highly individual (gymnastics, swimming). They will not sign a co-op for the more competitive sports in our area such as hockey, soccer, etc.