Cortisone shots help specific problems in joints, not just mask them. Its an anti-inflammatory, hella stronger than your OTC advil/asprin. Locallized injections will stop the recruitment of more inflammatory cells and also stop their release of inflammatory mediators. Since these mediators are what make you feel pain, yes they make the hurt go away. But progressive inflammation is also detrimental and can slow healing. If the damage to the joint is minor, the cortisone shot is a great way to speed up recovery. If its major, the masking of the pain will make you think everything is hunky dory but meanwhile, youve torn the shit out of a ligament or perforated a cartilage and the bones in your joints are going to feel some serious damage the next time you seriously load them.
Also, remember that cortisol will have many of the characteristics of hormones (like test). Prolonged systemic use will cause bloating and depress your immune system.
Dont think about shooting it yourself because something hurts. Get the joint checked out by a doc