Unless you are super wealthy (and the answer then is obviously, the person you hire), then the responsibility falls upon the couple to figure out how things will get done.
Thanks to the book I mentioned in last week's High Five for Friday, Adam and I found a list that helped us divvy up the responsibilities. Obviously, you don't have to do this along gender lines.
We started out by saying which household chore we personally hate most, and we both offered to do the chore the other hated. Then we distributed according to ability (helloooo, I know nothing about cars), and interest (Adam ADORES grocery shopping). From there, we just tried to be fair, and acknowledged that this is open to adjustment and change when needed.
The only alteration I would make to The Nest's list would be an option for "together," in addition to "him," "her," and "take turns." Most people's lives are busy, and if doing dishes together sometimes turns into a dance party, I'm not complaining.