Workers’ compensation laws vary by state, but generally, your employer (or their insurance carrier) has the right to select the treating physician. Of course, in the case of an emergency, you can visit whatever doctor you’d like. The other exception to this is if your employer refuses to provide you with a doctor’s name, at which point you are then able to go to a doctor of your choosing.
There is nothing preventing you from seeing your own doctor in addition to the workers’ compensation doctor. In fact, this is commonly done to get a second opinion in the event that the workers’ compensation doctor believes your injuries are less serious than you believe them to be.