Personal chefs have several clients and do menu planning, shopping and preparation of several days worth of meals. Including storage/freezing and re-heating instructions.
Private chefs work for one client or family.
You might want to check Careerbuilders or Monster. There's also personalchefs.com.
I saw a listing in Wisconsin that looked pretty cushy. (on the surface anyway.) $26k per year in addition to room and board all year to cook for a family from May to October when they vacationed. From October to May you were free to pursue your own thing.
So they do exist and I'm certain that there are even more lucrative positions out there.
Have fun.
April