What Authority do "Off-Duty" Police Officers have?

Its basically the title, what power does an "off-duty" police officer have? By this, I'm referring to some who work for the local police/sheriff's office but is not currently on the clock or out of uniform. I'm not referring to an officer who is working a second job like security or something because of Im aware that most stations allow their officers to retain their authority and gun while actively working these types of side jobs. 

An example being, I'm at a retail store or out somewhere in public, and for some reason or other, I come into contact with someone who flashes me their badge. Outside of the full rights that a civilian has, what other authority do they have? Do I have to lawfully follow orders they say or pay them any mind (outside of actual emergency situations or situations where someone can get hurt)?

As far as I am currently aware, they can just call for an officer who is currently on active duty to remedy a situation or in emergency cases pull over someone who is driving drunk or anything of that nature. I also know that if a off-duty officer acts without proper circumstances, they can have liability issues.