Just my opinion, but unless your by-laws prohibit it, it would be like filling any other elected position, except the two would run as a team. If they are co-presidents they would be sharing that position, so either taking turns (like running the meetings) or handling it jointly.
On responsibilities, either dividing them up or sharing them would seem to be the options. I would suggest if you need a vice president, elect the person as that to begin with.
There was another recent posting on this topic at
www.ptotoday.com/ubb/Forum24/HTML/000148.html,
so you may want to review those comments.
We had co-presidents at the beginning of last year. Although, some make it work really well, I'd never try it again. My co-president was wonderful, but it was just way too difficult logistically to coordinate the day to day decisions.