I can only say, judging from my own research, it is more difficult as an DO to get a competitive residency. When you do a search on the residency boards, the only time a poster posts specifically wondering how much scrutiny their DO degree would get, it's usually in one of the more competitive residency forums.
If you are 'only' aiming for a 'competitive' residency (rad, derm etc), it seems attending a top MD school would be best, and attending an MD school would be best over a DO school if you are hoping to do residency in the east coast and it seems, in California (despite their two DO schools?!). That's the impression I got from what I've seen and read.
Of course, some of this could be just wishful thinking on the part of some MD students---I've noticed that there is agreement that your MD school won't matter, just your degree, yet some tend to think DO students will get kicked to the curb---all of which seems to make no sense: if a PD is willing to discriminate based on your school (MD/DO), why wouldn't they also discriminate based on the top MD schools vs. the bottom MD schools? So in a nutshell, my personal belief is that go to the best med school you can, schools like Harvard and Hopkins will probably help a lot more than your DO or state MD school.
However, if you don't mind tending to primary care or the 'less' competitive residences like ER, then any MD or DO school would be good enough.