The short, uncomplicated version of the answer is the confusion over the use of the initials "DO". Without drudging through too much history and what-not, DO stands for two very different things. In the US, D.O. is a Doctor of Osteopathic Medicine - a physician with full practice rights. In most of the rest of the world, D.O. is a Diplomat of Osteopathy - essentially another term for a chiropractor or a clinician who can ONLY implement manual/manipulative medicine and has no prescriptive or surgical authority. Hence the confusion...
Realistically, the granting of practice rights of any Doc trained in one country who wishes to practice in another ranges from relatively simplistic to detailed and complicated to say the least. If you, as a US trained physician (DO or MD), wish to practice in another country, you have to apply/petition that country's medical licensing authority for practice rights -- this involves, among many other things, substantiating the training and education that you received.
While I have not done this...I suspect it is not easy for either an MD or a DO; but, due to the lack of confusion due to the initials...the MD probably has less of a time getting practice rights granted. However, since you medical authority is vested in your training and not just the alphabet soup behind your name, with some perserverance, I would imagine you could obtain those same rights if you are willing to jump through some additional hoops.
We can certainly thank the wisdom of the AOA for their white-knuckled death-grip on the past for the persistence of this confusion. Most reasonable, rational bodies would recognize that if a few billion people <rest of the world> see "DO" to have one meaning (esp a meaning that we'd prefer not to be associated with the profession) and only 300 million (United States) folks see "DO" as something entirely different -- that it might just be reasonable, logical and prudent to CHANGE (the treachery & treason of such a suggestion!) our designation such that the confusion, both domestic and international, is ameliorated. Yes, there might be some initial additional confusion here in the US, but in the long run, clarity would prevail for a much larger audience.