Private Treaty means that the seller does not want to display the price publicly and may only release it to serious inquiries. You will need to contact the seller directly to inquire about the price.
Private treaty is just another way of saying that the seller communicates and negotiates directly with the horse buyer. Private treaty sales are those that happen directly between the buyer and the seller. The seller has more control over the price and the home the horse ends up in.
Another possible use for the Private Treaty label is when marking a horse Sold or Pending Sale, Private Treaty can be used to mask the sales price from public view.