You MUST take your gloves off in a fight. Leaving them on signals you don't want to fight and if your opponent hits you he will get extra penalty minutes. If, on the other hand, you punch with your gloves ON, you will be given a fighting major, a game misconduct, and probably a suspension.
So, when a player drops a stick most often you will see them not pick it up and make sure they are in the proper defensive position. They will simply do what they can until a teammate can either clear the defensive zone or ice the puck. Downloading or using the image without the owner's consent is prohibited.