Private investigators are most commonly not law enforcement unless they are working for law enforcement to do some private investigating for the law enforcement.
If it's just a private investigator hired by someone to investigate someone for them and follow that person around then those private investigators have no authority to make arrests or are working for law enforcement.
So those people are just like you and me and can go across state lines to investigate someone if they want too or need too.
Some private investigators if paid enough can and will cross state lines to perform their investigations and yes they can legally cross state lines to do the investigations.