"I work in the private sector."
Translation:Je travaille dans le secteur privé.
I think you're right that it should be accepted, though I don't think "le secteur privé" is at all wrong.
My understanding is that while "le privé" can mean "the private sector" if the context is clear (as it arguably would be here), "dans le privé" can also mean "in private life". Does that accord with your understanding?