They pretty much have a monopoly unless you've been out longer than a year. Here are some relevant passages from the NRMP website:
A U.S. senior is a fourth-year medical student in an LCME-accredited U.S. allopathic school of medicine. A student with a graduation date after September 1 in the year before the match is considered a U.S. senior. U.S. seniors are sponsored by their medical schools in the NRMP Main Residency Match. Information about the results of the Match is sent to the U.S. senior's school of medicine.
If a senior student is withdrawn from the Matching Program because he/she will be ineligible to enter graduate medical education on July 1 in the year of the Match, the student may accept a position outside the Matching Program provided training will begin no later than February 1 in the year following the Match; however, if the student elects to participate in the Match, he/she will be a sponsored applicant.
I'm a US senior in an allopathic medical school and I've been offered a position outside the Match. What should I do?
If any of a teaching hospital's graduate medical education programs participates in the NRMP's Main Match, all of the hospital's programs may accept US seniors in allopathic medical schools only through the NRMP or another national matching plan. Offering a contract outside the Match is a violation of the Match Participation Agreement. US allopathic seniors who obtain PGY-2 positions through another national matching plan must obtain their PGY-1 preliminary positions through
the NRMP's Main Match.