You do not do anything.  Private listings are set up by the seller and your ID will show as Private to other bidders.  Only you and the seller can see your full ID.

just proceed as if it were a regular auction. Bid the amount that you are happy to pay if you win, cross your fingers that no one wants it more, and if you're the lucky winner, pay and await the arrival of your package in the mail!

Best of luck! I hope that you win!