Yes, anyone in the Circle who desires the item on the blanket can add their item(s). A recipient is selected from all those offering to trade. You can also add things to your initial offer to make it more enticing.
Another very important thing is no "hard" feelings are allowed. Everyone must respect the trader's choice for a recipient, or if no trade is accepted and the item is withdrawn. That can be done as well.
One lady offered a very unique buffalo hide bag. The flap was covered with thick buffalo hair and reminded me of Cousin It on The Addams Family. A lovely leather vest was placed on the blanket, and I offered my vintage fringed vest, not as nice. I just knew my offer was "trumped" when the leather bolero jacket went down on the blanket. For whatever reason I was chosen and honored. That trade was finalized with a hug!
Now this caught me by surprise, but I was there to learn. Later in the Trade Blanket, the lady offered up the fringed vest she had traded with me. One might think that a bit rude. Why would you trade for something and then try to get rid of it? I remembered the eagle plaque. I had followed my heart in that trade, as I really didn't need the item received.
The lovely lady had followed her heart in our trade as well. Perhaps it was to make my first Trade Blanket experience a success. Or maybe that very unique, crazy-looking bag needed to be with someone equally unique and crazy