First of all, the "training styles" for DOs and MDs is not all that different. In a DO school there are osteopathic principles which are stressed throughout the education, but when you think logically about these principles, they are not "osteopathic minded"...they are really principles that make a good physician.
Look at the patient as a whole.
Parts of the body are interconnected.
When there is something wrong in one part of the body, other parts of the body can be effected.
If the body is in 'perfect' working order, there should not be disease.
We can maintain and promote health by maintaining the bodys natural homeostasis and natural working order.
These are not things that only DOs practice, believe, or see when they look at a patient. Ask most any physician if they feel these things to be true and I bet you they would agree.
As far as the MD and DO degrees, I really dont see the point. There are cases where MDs from other countries have obtained a US DO degree (NYCOM program, for example) and now practice here. There are cases where DOs can literally buy an MD degree from offshore medical schools, but the degree confers you no additional benefits. From those I have talked to, this 'on paper MD degree' is, more often than not, looked down upon in the medical community. Lay persons may not care nor know what is means, but surely you tell an educated person you went to a DO school then got your MD degree by taking online courses at XYZ Carib med school...well, you fill in the blanks.
So why have both degrees? If you feel you are inadequate with the letters behind your name. If you want to put 'MD' on a sign outside your office. If you have the money to spend on it. Other than that, I cant think of a good reason.