I think there is no tag to specify a “private road”: the decreasing levels of importance of highways are primary / secondary / tertiary / residential. But a lot of streets are private, which should be below “residential”.
“Private” would mean with restricted access, for instance with a gate at the entrance.
Would such a tag be redundant with any other?
(and by the way, to link this discussion with a previous one about parkings with entrance and exit on distinct roads, a parking could be specified as a “private road”, the idea being that nobody would consider it as a shortcut.)