Hi.
Yes, a private auction will only allow the seller to see the IDs. It remains hidden, even after completion. You as the seller can see all of the bidding IDs at all times.
When feedback is exchanged, the IDs are shown and the auction is hidden as Private where there is usually a link to view the item. Something to think about if you only have a very few private, it's easy to use the completed option for someone trying to track another's purchases. For trust, you'll need to guard their IDs well.
Keep in mind there are many people totally opposed to private listings and would never bid on them. This may seem obvious, but just in case, I would say choose very selectively, i.e. don't miss your target market by going private. Personally, my buyers that prefer privacy are some of the absolute best I could ask for!
Hope this helps. Welcome to the Auctiva community!