Yes private investigators just like anyone else can legally carry a gun whether it's loaded or not as long as they are legally able to carry the gun and have a permit for the gun when needed.
There's no law against a private investigator not carrying a gun just like anyone else can carry a gun as long as they have no convictions that ban them from carrying a gun.
Most private investigators will indeed carry a concealed gun just in case they need to use it however most times the private investigator will never need to pull the gun.
So be aware that if a private investigator is following you that they could have a gun on them but they may not always have a gun but most do.
Most private investigators are just individuals without any powers other than to just follow you although a few private investigators may work for law enforcement or be an actual police officer in disguise to investigate and gather evidence for a case or to track someone down.