To generate processive motion along a polymer track requires that motor proteins couple their ATP hydrolysis cycle with conformational changes in their structural subunits. Numerous experimental and theoretical efforts have been devoted to establishing how this chemomechanical coupling occurs. However, most processive motors function as dimers. Therefore a full understanding of the motor's performance also requires knowledge of the coordination between the chemomechanical cycles of the two heads. We consider a general two-headed model for cytoplasmic dynein that is built from experimental measurements on the chemomechanical states of monomeric dynein. We explore different possible scenarios of coordination that simultaneously satisfy two main requirements of the dimeric protein: high processivity (long run length) and high motor velocity (fast ATP turnover). To demonstrate the interplay between these requirements and the necessity for coordination, we first develop and analyze a simple mechanical model for the force-induced stepping in the absence of ATP. Next we use a simplified model of dimeric dynein's chemomechanical cycle to establish the kinetic rules that must be satisfied for the model to be consistent with recent data for the motor's performance from single molecule experiments. Finally, we use the results of these investigations to develop a full model for dimeric dynein's chemomechanical cycle and analyze this model to make experimentally testable predictions.