Is it a public school?
Yes. Charter schools are prohibited from charging tuition or discriminating in enrollment. The school must meet all applicable health, safety, and civil rights requirements. They are subject to financial audits in the same manner as a school district. They are subject to the same guidelines on teaching religious doctrine as other public schools. They must answer to either the local or state school board for their performance, and they must issue annual reports, available to the public, that will include a financial statement, the progress in reaching academic goals, and a measure of parental satisfaction.
Furthermore, if the school violates its charter, it will not be allowed to continue to operate. If the school does not perform to the satisfaction of parents, it will lose students (and the money that follows the child) and will either have to improve its performance (to attract more students) or close. This is more accountability to the public – and to parents – than most other public schools.