Good question. I'd say if it's visible on the aerial imagery and represents a path on the ground, then it's fine to map it.

If it's obviously public or private then you can update the access tag to show that. If it's not obvious, the access tag can be left empty for now.

If it looks like it's probably intended just to be used by one house or maybe a few houses, I think it's also fine to set the access tag to private.