Sometimes it is private. If it's the parking lot and there is a collision between private cars, it is quasi-public. If it is somebody on the property, restaurant or store, to buy something it is quasi-public. This is not at all clear. The corporation exists by the Constitutional authority of the FedGov, and may be owned, which is hardly the description of a private person.
It doesn't matter whether it's Joe's General Store (a sole propritorship) or Walmart (a corporation). Either one is a private entity as opposed to a state entity. Neither should be able to take the property of the other by way of state confiscation.
Notwithstanding who issues a corporte charter, it is more than clear to me that the constitutional purpose of eminent domain is not to allow property seizures to support corporate enterprise.