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When a road sign says "private road", this means that it is an unadopted, privately owned road, not that access is generally forbidden. For an un-gated privately owned road, the tagging you probably want is:

ownership=private
access=destination

Using access=destination tells routing software that it can be reached for deliveries, taxis, visitors, etc. but that there's no legal right of way and it's not a through route.

If a road is gated and visitors can only reach the gate without explicit permission (and someone opening the gate), then the gate and everything behind it can be tagged as access=private.

https://wiki.openstreetmap.org/wiki/Tag:ownership%3Dprivate
https://wiki.openstreetmap.org/wiki/Tag:access%3Ddestination