Defining your media as "Private" in the previous version of MediaSpace was an imperfect way of making your media secret. It allowed you to upload media and make it undiscoverable in MediaSpace while still being able to share a link to it from off the site. But if that link was discovered, then your media would become public to the world.
The new MediaSpace allows you the ability to make your media more public or much more secure. But it will not allow you to make media private in the same way that the old MediaSpace did. Unlike the previous method of sharing private media, if you upload media to MediaSpace and leave it unpublished, but then share a link to it, those who follow that link will not be able to access your media.
One alternative available to you is to embed your media on an external page - Drupal, Wordpress, Blackboard, etc. - rather than a link. This will allow anyone with access to that external page to view that media. The access control to this media then shifts away from MediaSpace and onto the external web page (Blackboard, for instance).
Another alternative is to assign your media to the Private channel. Only ONID authenticated users can browse content of the Private channel. The caveat being that every ONID authenticated user can browse the content of the Private channel. But it also allows you to share a link to your media, even with those who are not ONID authenticated.
Lastly, if you know precisely with whom you'd like to share your media, you can create your own channel, set the permissions to Private, and populate the members list with only the ONID names of those who should have access to that media. That access list can be as short as one or two people, or as many as every ONID user on campus. Note that this only works with users who have ONID accounts. Thus, non-OSU users would not qualify in this scenario.