What a surprise..... !!
Private auctions.... Since ebay hides bidder names, there is only one reason I can conjure up to run a private auction... IMHO that reason is to hide the item when the feedback is posted, as the full bidder name is shown there. IMHO it is to prevent other ebayers from contacting the buyer about the item they purchased once they are able to see who the bidder was, as there are some on ebay who will let a buyer know they have been, shall we say "sold a bill of goods". With the proliferation of Paypal and the ability to easily claim for charge backs by a buyer for SNAD (Significantly Not As Described) like a Lalique claim that is not accurate, my experience is that sellers that use that option typically, at least to my observation, have a large number of attributions on merchandise that are questionable (I am being kind). As soon as feedback is posted, anyone can contact the buyer direct, except for the rare occasion where ebay gives you the message that they will not allow it because you are not "dealing" with that user.
Craig