Don't know about Beefmasters. However, with Longhorns, unless the animal is exceptional and the bidding gets serious, probably can sell for more via Private Treaty.
Our experience with attending special breed sales is that the bidding can go in any direction, depending on sale location, auctioneer, time of year, and the quality of the people bidding. We've seen some fantastic Longhorns going for next to nothing at one sale that would have brought excellent prices at another special auction. Considering travel expenses, consignment fees, etc., unless you can double your price at an auction, Private Treaty probably most profitable.
On the positive side, the special sales are excellent place to showcase your stock, meet prospective buyers, as well as get quick cash for your cattle. And, you can always "P.O." (pull out) your consignment if bidding doesn't go the way you want (you'll still pay fees for consigning).