Note that when I am doing traffic direction with appropriate authority, most drivers will never have heard of the organization I'm a part of (a local CERT team), nor have any real way to vet the authority I have. The authority they can determine I have is mostly that I'm doing it and no police officer has come by to tell me to stop doing it.
(Note that CERT members do not and should not self-deploy, mind you, and if you choose not to heed our directions, we almost certainly can find a cop car to come let you know you screwed up.)
Also bear in mind, many people directing traffic aren't public safety personnel at all. Construction workers are private employees, events have their own staff which direct traffic in various contexts, including public roads.
There's no definitive way to determine that someone is authorized to direct traffic, cars just need to obey them.