Question: Which one is correct? Or are both acceptable?
\*He earned an MD and a gold medal from St James for his dissertation. 
\*He earned an MD and gold medal from St James for his dissertation. 

\*He was appointed Surgeon at Hospital A and Hospital B.
\*He was appointed Surgeon at Hospital A and at Hospital B.

Answer: 
In the second example, the second *at* could imply a separation of the two functions. 

If the consultant surgeon post was a single employment that covered both hospitals, it would be omitted, whereas it would be included if there were two separate appointments.
In the first example I would go for "He earned an MD and a gold medal from St Andrews", because to leave out the second "a" implies that the gold medal is an enhancement to the MD in the sense of "DSO and bar".

In the second example, however, "He was appointed Consultant Surgeon at Hospital A and Hospital B" is fine, because the first "at" functions in linking both hospitals with "he was appointed Consultant Surgeon".

However, you'd keep both "at"s in if you were going to extend the sentence to say something different about what he was appointed to in Hospital B - "He was appointed Consultant Surgeon at Hospital A, and at Hospital B he was recognised for long service".