Sometimes private investigators are hired by law enforcement to carry out investigations and those private investigators could actually be law enforcement and have authority.
However the private investigators that work for the public and are hired by the public to follow someone and investigate things or people are just regular people that are non law enforcement.
Those private investigators have no authority to arrest someone or any other type of authority and must follow the same rules and laws that other people do.
Most private investigators will indeed follow the laws rules and regulations so they do not break the law and get into trouble.