I interpret a lot as part of my job and here's what I've noticed:
Interpreting sucks, there's so much subtlety that is lost when interpreting. When a doctor asks our patients "what kind of pain is it?" and the patient describes it, if an interpreter didn't have a pre-med background or a genuine interest the difference between "pain from surgical scar" and "pain like before surgery" could be lost, as one example. I find too often that my colleagues or me are helping doctors come to conclusions by trying to make suggestions to fill in the gaps in communication.
On the other hand, it happens very often, practically at every appointment and everyday, that a patient and doctor "communicate" with body language. The doctor will ask a question and parents will respond before we translate anything. Or the opposite happens where a patient replies but the doctor just says "okay, I got that."
A couple of my friends had to train with patients who didn't speak their language and without an interpreter too. They joke all the time about their devised ways of taking histories. ("Heart problem in family?" in charades, anyone?) Somehow, they got it to work.
In light of this, I think in terms of practical application, fluency is far more important. For me, that is Spanish. However, learning a few words in several languages is also useful as far as establishing trust with your patients. Knowing how to say "Hello", "How are you?", "Thank you", etc will be appreciated. I work with a cardiologist who knows a few words in Arabic and never fails to impress a family. He says his hi and hello so well some families believe he is fluent. It's a great ice breaker and definitely helps the families feel more at ease.