Osteoarthritis is arthritis. This is simply when there is pain and inflammation due to the breakdown of joint cartilage and the underlying bone. It isn't a disease per se, but a condition. As a result, it won't "spread." Of course, it is common that if there is that kind of wear on one joint, it could happen to the other joint. It is common for people to get arthritis is both knees, for example.
Rheumatoid arthritis (RA) is a long-term autoimmune disorder that primarily affects joints. This is the other kind of arthritis and, yes, because it stems from an abnormal immune response to normal parts of your own body, it often does spread and can affect not only joints but skin, vital organs, heart and blood vessels.