In 1982 the U.S. Supreme Court held that the Equal Protection Clause of the Fourteenth Amendment was violated by a Texas law that authorized public school districts to deny enrollment to children not "legally admitted" into the United States. That holding, in Plyler v. Doe, 457 U.S. 292 (1982), was dependent on the Court's conclusions that illegal aliens are "persons" who may claim the benefit of the Equal Protection Clause; that no substantial interest of the State of Texas was furthered by discriminating against children who had no control over their parents' conduct nor their own undocumented status, but who would be disadvantaged for a lifetime if denied a public education; and that this law was not an effective means of dealing with the State's interest in preserving its limited educational resources because prohibiting employment of illegal aliens presented a better alternative to dealing with an influx of illegal immigrants.
Plyler is still controlling law today for all states. That case begs the question of the difference between immigrant and non-immigrant students.
There is no obligation of public schools to provide a tuition-free education to non-immigrant students. The most common categories of non-immigrant students are foreign exchange students (typically holders of J-1 visas), non-exchange students (F-1 visa holders) living with relatives other than parents, and dependents of non-immigrant adults in this country for a few months to a few years to work temporarily (visas include B-2, M-2, etc.). The common thread for the examples given is that the students hold visas. To obtain a visa, the student must sign an application stating that s/he is a resident of a country other than the U.S. and intends to return to that other country.
On the other hand, immigrants (adults and children) come to the U.S. with the intention of making this country their permanent home. Other than for occasional visits, immigrants have no intention of returning to their countries of origin. Schools may not question immigrant students as to their "legal" status and may not demand their "documentation." Pursuant to Plyler v. Doe, public school districts shall provide these students, assuming they meet residency requirements, with tuition-free educations.