It is plain that the interpreter has to be able to follow the conversation between the doctor and the patient intelligently. The real question might be "Should medical knowledge be the overriding determinant of an interpreter's ability to work effectively in a medical setting?" To answer that question, we need to understand what skills are crucial for the interpreter to do a good job.
To follow the conversation intelligently requires knowing the topic rather than just the terminology. Generally it means the interpreter needs to know the symptoms, causes, and the course of development of the disease; diagnostic tests, their principles, administration and risks; the treatment options, their pros and cons; as well as management of the condition.
A sizable portion of the population knows all these things about hypertension or diabetes without going through medical school. All that is required to pick up such knowledge is the skill to search and learn independently. The interpreter deals only with information transferred between the doctor and his patient, not between medical professionals having an in-depth academic discussion. A medical or nursing degree would be overkill. It is neither necessary nor possible for each interpreter to have a degree in everything under the sun.
So what are the really crucial skills? You will note the great majority have got nothing to do with medical school.
Excellent memory and note-taking skills to capture all the details;
Full and accurate transfer of the message in an idiomatic fashion without addition, omission, alteration, simplification and expansion;
Deep knowledge of both languages to convey the way of delivery, including intent, connotations and nuances;
Commitment and skill to intervene appropriately where necessary to maintain the channel of communication e.g. seeking clarification and repetition;
Absolute impartiality, not offering opinions or advice even when solicited, leaving the ownership of the matter firmly with the conversing parties;
No role ambiguity, which will be difficult if the interpreter is a practitioner of the profession, or even worse, a staff member of the user agency;
Regular professional development to extend their skills and have access to professional support.
A couple of centuries ago, the surgeon’s task was performed by barbers. Knowing how to use the razor was helpful, but grossly inadequate on its own. In the same way, having detailed knowledge about the subject matter helps, but it takes proper training in interpreting to interpret adequately without putting people at risk.