private eye vs detective

private eye and detective both are nouns.

Word	Noun	Adjective	Verb	Adverb
private eye	Yes	No	No	No
detective	Yes	No	No	No

As nouns, detective is a hypernym of private eye; that is, detective is a word with a broader meaning than private eye:

private eye: someone who can be employed as a detective to collect information and assist in criminal investigations
detective: an investigator engaged or employed in obtaining information not easily available to the public

private eye (noun)	detective (noun)
someone who can be employed as a detective to collect information and assist in criminal investigations	a police officer who investigates crimes
	an investigator engaged or employed in obtaining information not easily available to the public