Yes, we are doctors! Chiropractors go through rigorous training in their schooling, which is actually longer than medical school. Educational requirement for Chiropractors are among the most stringent of any health care profession.
Chiropractors most often go to school for 10 semesters, which medical doctors only attend 8 semesters. The course work is mostly identical, just the emphasis is different. Chiropractors focus most on the natural aspects of how the body can heal and how to treat the body with their hands, versus using medication and surgery as medical providers do.
Chiropractors have to pass a 4 part National Board exam, and the last part is a live practical were they have to diagnose the ailment of a patient with a doctor proctoring their work. Once the National Board is passed, then you are able to apply for your state license and pass whatever requirements that that particular state requires. And each state has its own continuing education requirements to maintain the state license.
Here is a link to the Missouri Board of Chiropractic Examiners to read about the requirements to have a profession Chiropractic license in the state of Missouri.