California is governed by the California Interscholastic Federation (CIF) and the primary rules regarding recruiting and transferring are in Bylaw 510. Additional rules associated with transferring where a connection to a former high school or travel team coach exists are given in Series 206 and 207. The rules apply to both private and public schools.
Basically the rule is that no school may provide "undue influence", defined as any act, gesture or communication (including accepting material or financial inducement to attend a CIF-member school for the purpose of engaging in CIF competition regardless of the source) which is performed personally, or through another, which may be objectively seen as an inducement, or part of a process of inducing a student,
It is extremely difficult to establish that undue influence has occurred. I live in the same area as Truman, and my son attended a private high school (in the same league as his) for pretty much the same reasons. The local baseball board has periodic bouts of allegations of misbehavior by the private schools, although ironically the rules are written sufficiently broadly that IMO any public school within 50 miles of a private is very likely to be in technical violation.
Anyway, frequently privates do relax their academic requirements for excellent athletes, and this is legal. At my son's school, financial aid to any student is based on need, and students receiving aid have to work before or after school in partial compensation for the aid.