I'm thinking all shelters must be different. They love anything you send; I just think if you ask them which they prefer, they'll get more use out the items. I did see a few puffy beds out in the waiting area and in the hallways so they uses both but the flatter ones are easier to wash and dry. Ours can't wash and dry the puffy ones in the machines they have.
Now my dog won't sleep on anything flat--the puffier the better altho he prefers a our bed to anything LOL>