The pressing cloth I bought from JoAnn's is very thin.
I think a pressing cloth is used to prevent "shine" on the fabric from the iron. Shine does not develop on quilting cotton fabrics, so the only reason to use a pressing cloth on cotton would be to protect against scorching if you have a very hot iron. Fabrics with other-than-cotton content, and fabrics with a nap, can look funny if pressed directly with an iron.
You can always test a piece of fabric with and without a pressing cloth to see if it is needed.