RE: Doe shoot by me and another hunter, Who's is it?
In NY and CT it's the person that drops the deer. Period, bow or gun. This is the legal precedent. During bow season, ethics and course training say the first lethal shot claim the deer---but that's still a hunter's choice, not the law.
Sorry but IMO his deer, period. Next time, as previously mentioned, try to break a shoulder or put one in the neck. They don't run far without a spinal cord.
I am quite confused by the DNR official's stance and demeanor however. In my opinion, as long as both parties were honest (and it sounds as if you both were), then it should have been a clearcut case. He should have informed you both that legally the other guy "owned" the harvest, but ethically it should be yours. Check OH games laws for definitions, but I'm quite confident that most states hold the same as NY & CT.
S&R