To be a private road system you would have to inhibit access by the general public with a gated system or some sort of security Once a road or trail is available to one or more persons it becomes for all intents and purposes a public conveyance. Now that has no bearing on who maintains the road but for the purposes of law, the public can access a road, trail driveway that is used for one or more people for travel. A sign is merely symbolic and not a revocation of rights since the trespasser can argue around that in many ways.
When it is your property and you have one driveway that is private, public or police cannot access unless permission is granted. Put up a gate. Doenst matter if its open and inoperable.