A personal chef is essentially a professional chef who prepares meals for a customer or a family. This can happen in the customer's home or in the chef's own kitchen. A private chef, on the other hand, is a chef who works in a private home, usually for a single client or family. A personal chef is a chef who is hired by different clients and prepares meals in the kitchens of clients' homes, according to their needs and preferences.
A private chef works for a single family, an individual or a couple. Someone has hired them, paid them a salary and will most likely pay them benefits as well. They have fixed schedules or perhaps flexible schedules.