In most investigative games, a GM should want the players to get their hands on as many clues as possible - the challenge for the players is how they interpret those clues and face the dangers they lead to. Because of that, I try to let skills be wide and have plenty of overlap when used for investigation, but I take a more narrow approach when they're being used to face danger.
Profession skills are intended to overlap, but only in the context of the profession. A race car driver can use their skill instead of drive - if their on the track. If they're trying to get a truck full of explosives up a muddy hill, Profession: Automobile Racer wouldn't be useful, but Profession: Trucker would. The drive skill could be used in either place, but it couldn't be used to understand the business of professional athletics like Profession: Automobile Racer could.
I'm personally opposed to skills that serve as general "lie detectors", like read people or psychology. To me, deciding if you find someone's words believable based on context is a big part of the game. I would allow Profession skills to tell a fake, but only in the context of subjects where they have expertise. For instance, I'd let a musician roll their profession skill to see through a record producer's lies in a business deal, or I'd let a soldier roll to realize someone is faking having military experience. For a private sleuth, I might let them roll their profession skill in the context of catching a cheating spouse or an industrial spy, which tend to be the two sorts of jobs that private detectives end up working.