store detective vs private eye

store detective and private eye both are nouns.

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

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

store detective: a private detective employed by a merchant to stop pilferage
private eye: someone who can be employed as a detective to collect information and assist in criminal investigations

Other hypernyms of store detective include operative, PI, private detective, private investigator, shamus, sherlock.

store detective (noun)	private eye (noun)
a private detective employed by a merchant to stop pilferage	someone who can be employed as a detective to collect information and assist in criminal investigations