Private bidders set the fact that they want to be Private in their Profile. Note that this is NOT the same as a seller declaring an auction Private. (ebay's words are in italics)

Here's what we found for "How do I set my Feedback Profile public or private?"

You can make your Feedback Profile public or private. By default, it’s set to public.

When your Feedback Profile is public, you increase the chances that other members will do business with you. It’s a good way to build trust.

If you make your Feedback profile private, you won't be able to sell items on eBay. Your Feedback comments will be hidden, but members will still be able to see the number of positive, neutral and negative Feedback ratings you've received.

This is old wording! With the new ebay regulations, a bidder can't receive anything but positive feedback, but you could see any comments a seller made about your transaction as a bidder.

Now, what is a Private Listing? Here's what ebay says:

Here's what we found for "How private listings work"

You can create a private auction-style listing to keep your buyers' identities anonymous. This hides usernames and email addresses during the auction and after the listing ends. Only you and the winning bidder see the listing results, and only you know who bought the item.

How to create a private listing:

When creating your listing, select Switch to advanced tool.
Go to Choose a format and price, then select Add or remove options.
Select Private listing > Save.

As you can see, the choice to get involved in anything "Private" is up to the bidder. He has to choose to make himselve a Private Bidder. And he can choose not to bid on any auction that the seller has declared Private (as I do).