The only real restriction is you as an individual. Most of the people who say otherwise have not been through the process themselves (premeds or MS1/2). If you score very highly on COMLEX and interview well you can pretty much choose whatever residency you want (though not always where). There are DO's in many of the top programs in the country. If you plan on doing pain clinic anesthesia, orthopedics or PM/R at a top program DO degree may help you get noticed above the other 150 applicants, since your osteopathic skill really can set you apart.
If, however, you only went DO because you werent competetive enough to get into an MD school, it is much harder to get into some of the top allopathic residencies than it is to get into their medical schools. What makes you think you will fare better? Some of these programs require statistics at the undergrad equivalent of a 3.9gpa and a 40 MCAT score. Additionally, you should know someone who knows the program director personally- almost all the applicants who get accepted have connections like this. Do some out rotations. Also, it helps to have your name on several publications during your first 3 years of medical school. Many of the MD applicants have this. Get on it. These factors are FAR more important than whether you took the USMLE vs the COMLEX. You cant blame your degree for your own personal failures.
There is genuine anti- DO bias at a few backward podunk hospitals in states without DO schools. You may need to take the USMLE to get into one of these places, or they may even exclude you if you take the USMLE. Get over it. For every one like this there are 5 that wont treat your application any differently than the MD applicants and 1 that will prefer you over an MD because of your degree alone. Harvard, U of Wash, Cleveland Clinic and Mayo all openly welcome DO's to their most competetive programs, just to name a couple.
Also, there are several DO-only residencies in some of the most competitive specialties if you're numbers arent good enough for the best allopathic residencies in the country. There are something like 5 DO derm residents here in kirksville. There are several DO anesthesiology residents, several DO surgery residents. This is a very small town. You can get into DO derm or DO surgery residencies with even average numbers if you know the doc and have some charisma... a backdoor way into very high competition fields. I have heard many MD's complain about how unfair this is, and how its so much harder for them to get into competetive specialties than it is for DO's because of this.
The only real restriction with the DO degree is working internationally. This is only true for certain countries, and people are working on it, but it continues to be a challenge. There are many countries that welcome DO's however. If you arent too picky about where you go this isnt that big a deal.
when all is said and done though, ask yourself why you want to be a DO. If you dont have a answer you're happy with- go MD. If you do have a good answer- no restrictions will hinder you.