I have always removed the rings from my jars, whether pressure canned or water bathed. If you want to pretty them up for gifts, you can cut a circle out of fabric using pinking shears, lay it across the lid, and rescrew on the rings - loosely. A good way to store your rings (after washing and drying as chickhill noted), bend a metal clothes hangar into a loose circle, unwind the top twist and thread them on. You can re-attach the twisted end loosely and hang in a closet or other out-of-the-way spot until needed again.

I have also used the fabric circles. I use one of those loose mesh fruit bags w/ the drawstring closure for storing jar rings.

I never have them rust or stick but maybe because my Mom taught me to add some vinegar to the water in the canner since we have very hard water. Even though I have had a softener for years I still do it. Just about a couple of tsp. to the canner and when the jars are cooled, I remove the lids and dry the jars and rings.