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Your are getting into deeper waters there, aristo. "Public" and "private" are somewhat similar (the distinction is subtle), and the alternative is "state" school.
However, to answer your question, I think these would only be used where the noun is implied.
"What sort of school does your son go to?"
"Public."

Oh yes, I do realize that what you call public is the opposite of what it is here; I just meant using the adjective as a substantive. Thanks for your patience. Just trying to learn English.

They do say that some east coast American dialects have preserved English from Elizabethan times; it is us (or is it we?) who have diverged!