This rental actually does vent outside. Our first home vented outside and when we gutted to kitchen in the second one I insisted on a vent to the outside (ended up being easy as I placed the range within inches of an existing vent to the exterior so it just required a flexible connector to hook it up.).
Sometimes the vent runs through a soffit and people tear down the soffit for aethestic reasons. Then they end up installing a recirculating fan. And some places just have no vent to start with. Down drafts aren't the greatest vents anyway so you might not notice that much difference between recirculating and downdraft.