Besides the adult/pron type of auction, there is very little reason to have private auctions. I would not bid on private auctions because chances are HIGH that the seller may be shilling with a second/shill account.
You can't tell and you cannot check the other bidder's history.
I have read that sellers who deal in jewelry or in this example watches may have private auctions because it is a very competitive market. Their claim is that private auctions will prevent other sellers from contacting "their" bidders with low-ball offers or to bad-mouth the said seller. There are probably other reasons but a private auction will make me click the back button faster than you can say shill.