Where did Asian Americans go to school during the 'separate but equal' era?

Short Answer

This really depended on several factors, including state, city, time, Asian origin and even international politics. Local housing regulations also influenced the schools that Asian Americans attended. Asians were sometimes treated more favourable than African Americans, sometimes the same and sometimes even worse. Thus, options could include either a segregated or desegregated public or private school, or home schooling. In short, it's complicated.

Details

Chinese in mid to late 19th century California were confronted with discrimination to varying degrees. Children might be able to attend 'white' schools if the white parents didn't object, but local laws sometimes made no mention of Asian children, instead focusing on segregated schools for Indigenous and African Americans. Home schooling was one option for Asian Americans, private schools another. It was not until 1885 that the first Chinese public school for elementary ages was established.

Varying and frequently changing policies continued into the early 20th century:

During the early decades of the twentieth-century, school segregation policies affected Asian Americans differentially depending on their specific heritage, geographic locale, and the prevailing social climate. In 1900 89,863 people of Chinese birth or descent lived in the United States, most on the West Coast.

Source: US Dept. of the Interior, 'Racial Desegregation in Public Education in the United States' (2000)

California accounted for a large percentage of Asian-origin residents and even within this state policies varied. Note this interesting example:

Living with his family outside of Chinatown, Dr. Wong Him enrolled his daughter Katie in a neighborhood school, but a year later, Katie was instructed to attend the Chinese Primary School. The doctor filed suit noting that the San Francisco school board permitted blacks, American Indians, and Japanese to attend local schools while targeting only Chinese for discriminatory treatment. In Wong Him v. Callahan (1903), the U.S. District Court disagreed and upheld the idea of “separate but equal” as stipulated in the school code.

Source: US Dept. of the Interior

In 1905, the San Francisco School Board ruled that

Our children should not be placed in any position where their youthful impressions may be affected by association with pupils of the Mongolian race.

The discrimination against Chinese was soon also applied to others:

In response to the challenge of changing demographics more than a century ago, the San Francisco School Board established a segregated Chinese Primary School for Chinese children to attend, including those who were American-born. By the turn-of-the century after Japanese immigrants had settled in the wake of Chinese exclusion, the School Board also applied the Chinese segregation policy to Japanese students.

International politics then dictated a reversal of the decision to desegregate Japanese students:

Japanese American parents refused to comply, filed suit (Aoki v. Deane, 1907), and appealed to their homeland by reporting their mistreatment to Japanese newspapers and the consulate (Wollenberg, 1976). Interested in maintaining healthy relations with the Empire of Japan, President Theodore Roosevelt intervened on behalf of the Japanese and forced the school board to relent (Daniels, 1992; Wollenberg, 1976).

Source: Joy Ann Williamson, Lori Rhodes and Michael Dunson, 'A Selected History of Social Justice in Education'. In 'Review of Research in Education Vol. 31, Difference, Diversity, and Distinctiveness in Education and Learning' (2007)

However, segregation was not universally applied. The amended 1921 California school law stated

The governing body of a school district shall have power to exclude children of filthy or vicious habits, or children suffering from contagious or infectious diseases, and also to establish separate schools for Indian children and for children of Chinese, Japanese, or Mongolian parentage. When such schools are established, Indian children or children of Chinese, Japanese, or Mongolian parentage must not be admitted into any other school.

Cited in US Dept. of the Interior

However,

only four towns (Walnut Grove, Courtland, Florin, and Isleton) duly segregated Japanese and other Asian American youth into “oriental” schools, all four were small farming outposts in the San Joaquin River Delta of northern California.

Source: US Dept. of the Interior

In mid-1920s San Francisco, local authorities came under pressure from Italian Americans especially to build a secondary school for the increasing number of Chinese children. Francisco Junior High in North Beach, near Chinatown, did not accept Chinese students. Then,

After a year of contentious Board of Education meetings, Chinese American students won the right to enroll in Francisco largely because the city lacked the money to pay for a segregated junior high school in Chinatown.

Source: Charlotte Brooks, 'Alien Neighbors, Foreign Friends: Asian Americans, Housing, and the Transformation of Urban California' (2009)

Meanwhile, in poor areas in pre-WWII Los Angeles, Asian children might attend the same schools as Mexican, Greek, Italian, African and Jewish Americans. In more affluent areas, however, when Japanese Americans

placed their children in the public school system, white resistance mounted; ...If the Nisei* school-age population grew too large in such areas, white parents knew to invoke the state constitution, which allowed the segregation of Asian American students in separate classrooms or separate schools.

Source: Brooks

Nisei: US-born children of immigrants from Japan."

Information on the northern states and midwestern states is scarce, perhaps because the number Asians was low. Atlantic City High School in New Jersey appears to have accepted Asians as the writer George Kin Leung graduated from there in 1918. Segregation was the norm in the south but, again, laws were not exactly the same everywhere, and often existing rules or laws were challenged:

While laws pertaining to school segregation for Asian Americans varied throughout the South, battles between school districts and Asian American parents over their right to send their children to white schools resulted in local, state, and federal courts determining the “correct” race of Asian Americans for segregation purposes. Asian Americans did not readily accept the courts’ decisions, however, and actively pursued their rights to send their children to white schools through lawsuits and appeals.

There was, though, often disagreement among whites as to segregation policies concerning African and Asian Americans: some believed African Americans should be treated more favourably than Asians, others held the opposite view.

U.S. Supreme Court Justice John Marshall Harlan made a striking comment that reminds us of the role Asians and Asian Americans have always played in the U.S.’s otherwise binary, Black-white racial system. Justice Harlan’s famous dissent against the “separate but equal” doctrine in Plessy v. Ferguson (1896) contrasts the loyal and patriotic African American with the Chinese, a “race so different from our own that we do not permit those belonging to it to become citizens of the United States,” and goes on to lament how the Chinese could ride in a railcar in Louisiana with whites while Blacks remained segregated under force of law.2 Jim Crow segregation would regularly group Asians alongside whites, yet others like Harlan saw whites and African Americans as united against the intrusion of these “strangers from a different shore.”

Concerning Brown v. Board of Education,

In 1954, the Supreme Court declared racially segregated schools unconstitutional in its famous Brown v. Board of Education decision. African Americans initiated the case, but the decision affected all other racial and ethnic groups similarly situated.

Source: Joy Ann Williamson, Lori Rhodes and Michael Dunson

Although a landmark decision, it did not immediately change the situation, and some districts moved much more slowly than others. Further, as reirab points out in a comment below, the ruling did not apply to private schools. Thus, in 1976 the Supreme Court issued another ruling on segregation in schools. There was a

landmark decision in Runyon v. McCrary, ruling that even private, nonsectarian schools that denied admission to students on the basis of race violated federal civil rights laws.