I looked this up in my state. They are exempt on private roadways that are only open to select persons via implied or explicit permission[]. They do not appear to be exempt on private roadways that have public access, which is what all the roads around me are. I cannot even selectively limit access on my own roadway because it has an easement for the public to pass. But I still fully own it and am responsible for all of the maintenance.

Therefore it does appear the plates are required even though they are fully privately owned roads and privately maintained. Because our roads don't meet the definition of 'private' road in my state even though they're completely private.

Not legal advice.

[] https://www.azleg.gov/ars/28/00601.htm