The odds of somebody having the same bday as you would just be the 365 (or 365.25 if you want the leap year too) You wouldn't factor in the odds of your own bday. Thats also assuming an equal number of people are born every day, which isn't quite true, some days are much more popular than others. Its also not taking into account the odds that one of those people with your same bday would happen to be at the same place as you AND happen to know its your bday.
Since it's two people, which means you have to take 365,25 possible birtdays times 2?