Entanglement entropies of spatial regions are useful quantities for characterizing the ground states of quantum field theories. However, their usefulness depends partly on two assumptions: that they are universal and duality-invariant for a given theory, and that they distinguish between different theories. We will explore whether these assumptions are correct, focusing on two free two-dimensional conformal field theories, the Dirac fermion and the compact boson. Explicit calculations of entanglement entropies in the literature suggest that these quantities may fail to agree when calculated in theories that are known to be dual, and may also fail to distinguish between theories that are known not to be dual. While getting at the root of these puzzling discrepancies, we will discover a new kind of duality between these two well-studied theories.