Bats have hollow bones and are generally as light as they can be and still be able to hold out their wings. Dogs generally have pretty dense bodies, so I think their wings would have to be very large, and the larger they got, the heavier the body would be and the larger the wings would have to be, at least at current body density. Dogs would have to either lighten up considerably, or develop some kind of very, very strong muscles and connective tissue to support wings that would carry them.
If they could manage to solve these problems, I don’t know what prey they would go after, and I don’t know if they would remain pack animals. Perhaps they would become predators like eagles. The shape of their bodies would have to change to streamline them so they could fly at high speeds in order to attack large animals. They would probably need to have six appendages—four legs and two wings, so they could defend themselves while on the ground. Perhaps they would use their wings for defense, sweeping them around to knock other creatures away.
I think that if dogs could fly, they would have wings most like birds of prey. They would be able to glide for long distances and get very streamlines to attack their prey. I think it would be funny to see packs of flying mammals, but not so funny because if you were out in the open, you could be at risk.
If we could domesticate flying dogs… imagine the hunting! And kids trying to get the dogs to fly them. Rescue dogs would take on a whole different meaning, although they would probably have to have very different lungs to fly high.
If I had a flying dog, I might name him Red Rover.