If these additions should only be visible to the user that added them, then you cannot add the data to OSM. You will have to store that data using your own infrastructure (database servers etc.).
Note however that if your user adds a driveway based on the location of a (say) street that you displayed to her using OSM data, then the data entered by the user is subject to the OSM data license. In other words, you cannot claim your own ownership of that driveway data because it was derived from an OSM basemap or OSM vector data. As soon as you distribute (or show) that data to your users, you will have to release it under OSM's data license (with effectively renders it public).
In other words, I see a lot of legal problems with your proposed approach. I suggest taking a look at OSM's legal FAQ and asking this same question on OSM's forum