Parrots' beaks are REALLY strong because it's designed to crack hard nuts. A dremel would be overkill for a chicken's softer beak.
Just whip out a set of nail clippers and trim off the end a teensy bit at a time. A bird's beak has a spot in it, just like a dog's claw or your fingernail, where is stops being unfeeling tissue and starts having blood vessels and nerves. You'll probably be able to see light through the thinner edges of the beak, so trim part back that and it should be fine. When you're done, use a file to shape the beak edge.
Easy! This will take you a max of 5 mins.