Panamanians doctors offers their services in Panama either in private or public clinics. Due to government medical requirements, graduates must complete a mandatory two years of practice before they can receive their medical degree.
I assume that, in the question, when you say "American-trained" that you are referring to US training..?
Well, Panamanian medical graduates usually are very highly trained, not only in the US, but Europe or other countries on the American Continent, in medical schools are considered prime universities, making the medical staff in Panamá one of the most well respected by locals or foreigners of any place in the world.
Casco Viejo, being a part of Panamá City, doesn´t offer the service of hospitals or clinics in their specific area because Casco Viejo s mostly historical and touristic with a very small neighborhood, but medical services are just minutes away, with private and state provided modern ambulance service as well, 24 / 7.