At my shop we are looking at getting osteopathic recognition and I was tasked with doing it, so here is what I can tell you. Some of this may be wrong, I haven't looked at the requirements in a while. So don't take it as gospel, but it'll give you the general trend. This is for an IM residency.
To have osteopathic recognition you need to have at least 1 DO/year in the track averaged over 3 years. They get additional lectures and training in osteopathic principles and practice, such as participating in an OMT clinic. There is no formal research requirement other than faculty are expected to lecture and be leaders in their field, which most of us in academic settings already do. If our residents do DO related scholarly activity that's gravy, but their ACGME required scholarly activity otherwise fulfills any requirements I'm aware of.
MDs are able to participate in the osteopathic track and get additional training through it. Does this mean anything? Who knows - you don't need a special certification to perform or bill for OMT that I'm aware of so I don't see why an MD couldn't practice OMT. That being said, where it would become an issue is if there is a complication because the attorneys would have a field day with it. Now if your an MD who wanted to do OMT and have formal training through your residency then you would have a much more defensible position than "I attended a 2 day CME event".
Honestly the requirements to obtain osteopathic recognition are cake compared to general ACGME accreditation when you compare them. There are site visits but not for the first couple of years while you get a curriculum established.
What are the benefits to the program? Technically you could more easily recruit DO's, especially those looking to hone their OMT skills (this is a great biller in private practice).
As an MD should you apply? Absolutely. At my shop we are still 90% MD, but we figure if we can recruit a more competitive DO applicant because we have osteopathic recognition then it may be worth it.
Edit: from above replies, it appears some programs use this to keep themselves as "DO only" but that is a program specific ideal and not something stipulated in the osteopathic recognition guidelines.