Legal Question in Real Estate Law in California

It is likely that a leash law applicable in your jurisdiction does indeed apply to the private road. You need to read the actual ordinance for your locality (county, city).

I agree. A "private road" may be your personal property, or often it is a non-public facility jointly owned and used by neighbors. In the latter case, local ordinances will probably apply. In any event, and absent more information about the road ownership and rules, I'd be cautious about assuming that leash laws don't apply.