Would a dentist be both a DMD and a DDS?

No. Although the degrees are equivalent, the specific professional doctorate designated to qualify for licensure to practice the profession of dentistry, as well as seek further residency training in recognized advanced dental specialties (such as periodontics, oral & maxillofacial surgery, orthodontics, endodontics, oral pathology, pediatric dentistry, etc.), is based on a University's (and its School or College of Dentistry's) official authority to grant the said degree.

DMD = Doctor of Dental Medicine. DDS = Doctor of Dental Surgery.

The DMD is a relatively newer degree in the United States and has been conferred by more modern dental schools (established circa 1970 and after). In fact, several older dental schools, which previously granted the DDS, have converted to awarding the DMD (including Harvard University and more recently University of Illinois), while others maintain the traditional DDS degree programs.

In theory, the newer degree title emphasizes the role of the biomedical sciences and pharmacotherapy in the foundation of the curricula. Furthermore, the DMD title illustrates the increasing responsibility of the dental profession (as a field of medicine) in understanding, diagnosing, and managing systemic processes (the oral-systemic link), in addition to the surgical and chemical/material science approaches to oral rehabilitation. This concept is at least a century old, and applies to DDS curricula too.

It is likened to the "M.D vs D.O." comparison, whereby allopathic physicians vs osteopathic physicians, though different in name, sit for the same licensure examinations and compete for the same residency programs to practice various areas of medicine. However, they do not use the degrees interchangeably, nor in duality.

So, to answer the "DMD or DDS" question, one would refer to him or herself by the specific degree conferred, and shown to be awarded on the academic transcript, by the institution. Not both. If a lay person (non-healthcare provider) uses the titles interchangeably, that is one thing (and probably not a big deal), but it would still be inaccurate.

ONE FINAL NOTE: On the other hand, if one possesses a previously earned dental degree from a foreign country (e.g., a DDS from Colombia), and desires licensure to practice in the United States, he or she may be allowed to complete training in one the established foreign-trained dentist programs, which are offered at a number of U.S. dental schools. The programs typically consist of a two-year curriculum, whereby the foreign-trained doctors are updated on the techniques, philosophy, science and standards of care necessary to complete U.S. national board examinations and state licensure board exams (for particular states or U.S. territories). Completion of the program may well result in the award of a second DDS, or a DMD, in addition to their previous degree [which in theory could be used in duality (DDS,DDS or DDS,DMD) following their name].