Expats have a choice of either private (international) hospitals or public ones. Private hospitals are likely out of reach for those without private insurance since fees are on par with fees abroad. Public hospitals may not be the best choice for those without functional Chinese language skills; however, major public hospitals have VIP wings where you will be able to access foreign-trained, English-speaking doctors. Expats do use the VIP wings of public hospitals often enough that they are decently equipped to handle expat patients.
Western private doctors are usually part of a private hospital or clinic and are not in the practice of making house calls, although the Beijing Vista Clinic is an exception (check Hospital list for more information).
You’ll find that many doctors at private international hospitals/clinics are usually high-caliber professionals living in Beijing as expats themselves. There will also be a population of foreign-trained Chinese doctors on staff who may be experts in their field at Chinese institutions (and working part-time at private international clinics) with good English. Nursing staff, however, is almost exclusively Chinese but with good English-skills.