I've never seen a private buyer, but I know there is a certain point on items that are high dollar, that eBay hides the identity of the bidders to all but the seller. It is to discourage spam or inappropriate emails to the bidders. Those usually show as Bidder and a number. Your ID will show to you, but anyone elses will be hidden.
I hate to be outbid! If it is something I really want and I can, I will wait until the last few seconds to bid. If I can't be at the computer at ending time, I just keep my fingers crossed!