Plot hole: Throughout the entire show, the chief, Juliet, and Lassiter constantly tell Gus and Shawn to stay out of their way and off their case. The chief even reminds them several times that they can't work on a case unless she's assigned it to them, ignoring the fact that Shawn and Gus are private detectives and do not need police permission to investigate a case.
Suggested correction: But they're only involved in police cases by invitation. People can't just take it on themselves to wade into crime scenes, private detectives or not. Warnings like that might not strictly be a command they "have" to obey, but it's in their best interest to take it on board and not annoy the department too much.
Yes, but the detectives do not know whether they have been privately hired or not. If they had been hired, they have every right to be involved.
