As an applicant this year, I have no clue what my speciality may wind up being.. If I even get in, but I do know that global/international health is in my future.
That said, this seems to be a moot point for DOs because most of the best global health residencies and affiliate hospitals (NYU, Mt. Sinai, Yeshiva, Duke) don't accept DOs. If I know that I want international health.. Should I forgo my DO aspirations?
I'm not sure exactly what you are using for criteria as "best global health residencies" for a couple of reasons - first, there is no such thing as a residency in "global health". Family Medicine is a particularly good specialty choice for global health, but people from every field can work internationally and focus their careers on this field. Some residencies have global health *tracks* that can be interesting. Other programs may encourage international rotations. Other programs include didactics to help residents learn global health theories and concepts from a more academic perspective. Others tend to focus on training residents in broad spectrum care to have a skill set useful in underserved areas domestically or internationally. It's also relevant to consider what percentage of a residency's graduates actually practice internationally, full or part time. Because talking the talk is not the same as walking the walk.
All that said, I went to a DO school (Ohio University) and we had a robust international program with regular lectures and trips to Ecuador, El Salvador, Kenya, Honduras (used to have programs in China as well.) There's a chapter there of the AFMSA-IFMSA (International Federation of Medical Student Associations) through which students can do clinical exchanges in something like 100 different countries. A few years ago an OU student did an exchange to Palestine, which I noticed in your profile picture.
At the residency level, again, there are lots of ways to do international work. I'm at one of the "full spectrum" Family Medicine programs where tons of our graduates work all over the world, and I think this is some of the best training I could get to do that. This program happens to be ACGME accredited and it, like many other similar programs, is happy to welcome DO students.
I think one benefit of going DO if you plan to work in international settings is that you have an extra skill set to help care for people with musculoskeletal issues - I actually got interested in osteopathic medicine after working with a DO in Central America who had something more to offer than just celebrex to patients doing heavy physical labor daily.
Good luck to you, wherever you wind up!