You can share a Lightbox with particular users by clicking on the 'share with users' link on the Lightbox or Manage Lightboxes page.
The initial page gives you some information on the functionality and will display the list of users that currently have permission to view or edit the Lightbox through sharing. You can add/remove people from this page, as well as change whether they are able to edit the Lightbox.
The 'Add users' link will allow you to search for a user by Username, Email-Address or Group. Users may only search for other users in groups they are themselves in. Once you've selected your user(s) you can choose to send them a notification email, and also give them permission to edit the Lightbox.
Once a Lightbox is shared with a user, they will be able to see it in their list of Lightboxes. If the shared Lightbox is selected then it will work like a regular one, except that the user will only be able to add assets if they've been given edit permission on the Lightbox, and will only be able to remove assets that they have added. Users are only be able to see/download assets where they have permission to do so, even if the Lightbox is shared.