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.
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
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
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
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.
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.