public lavatory vs privy

public lavatory and privy both are nouns.

Word	Noun	Adjective	Verb	Adverb
public lavatory	Yes	No	No	No
privy	Yes	Yes	No	No

As nouns, privy is a hypernym of public lavatory; that is, privy is a word with a broader meaning than public lavatory:

Other hypernyms of public lavatory include facility, bathroom, can, john, lav, lavatory, toilet.

public lavatory (noun)	privy (noun)
a toilet that is available to the public	a small outbuilding with a bench having holes through which a user can defecate
	a room or building equipped with one or more toilets

public lavatory (adjective)	privy (adjective)
	(followed by `to') informed about something secret or not generally known
	hidden from general view or use