But isn't the information gained from a message necessarily subjective? It depends on just on this message, but from the distribution from which this message was drawn, and more appropriately, the distribution anticipated by the observer will determine the value (information content) of the message. That is exactly how it becomes useful in practice.

This "surprisal" can be parametrized by relative entropy (Kullback-Liebler divergence), which reduces to the Shannon entropy formula only when the observer has the correct anticipated distribution.