[sethg]

The public schools are obliged to take just about every kid who shows up, including children with severe emotional, cognitive, or physical disabilities. In the US, they are also obliged to provide disabled children with education “in the least restrictive environment”, meaning that if a disabled child can be educated with special accommodations alongside non-disabled kids, that’s what the school has to provide, even if putting the disabled child in a specialized classroom all day would be cheaper.

Private schools, by contrast, have no obligation to accommodate disabled students. For that matter, they’re more or less free to say “there’s nothing wrong with your child, but s/he wouldn’t fit in to our school”.

[patio11]

Private schools, by contrast, have no obligation to accommodate disabled students

That depends on your jurisdiction. For example, you've indicated you live in Massachusetts. Two minutes of Googling found that 151C (e) makes it unlawful for private schools to refuse admission to the blind, deaf, and students requiring use of dog guides. That is one law. I would not wager it is the extent of regulations on the matter.

[sethg]

A private school can’t refuse admission to a deaf student, but if (for example) the student requires a sign-language interpreter to participate in class, the private school is under no obligation to pay for one. A public school in a similar situation would have to hire an interpreter.