For centuries, there have been many risks attributed to entering fairy rings or picking the fungi. If you really must try one of these recipes, but really are concerned for your safety, try carrying thyme with you, or spreading it across the ring before picking. This is said to confuse the fairies and remove the risk they might pose without any lasting harm.
One definite risk is mis-identification. Marasmius oreades, although by far the most common, is not the only fungus to grow in a ring formation. Notably, a couple of the clitocybes are toxic and must be avoided (clitocybes always sound to me like an alien race of murderous, rapacious females, probably in a David Cronenberg film). Until you are sure of what you’re looking for, always carry a good guide book, with photographs, not drawings.
Fairy Ring mushrooms are easily dried, after removing the woody, inedible stalks, and can be stored whole or powdered. In his Family Cook Book, Hugh Fearnley-Whittingstall says they pickle well. Haven’t tried it myself, but all edible fungi seem to be good for pickling. Just make sure you use a recipe which jars them in oil, not brine.