We study differential Shannon entropies determined from position-space quantum probability densities in a coupled electron-nuclear system. In calculating electronic and nuclear entropies, one gains information about the localization of the respective particles and also about the correlation between them. For Born-Oppenheimer dynamics, the correlation decreases at times when the wave packet reaches the classical turning points of its motion. If a strong non-adiabtic coupling is present, leading to a large population transfer between different electronic states, the electronic entropy is approximately constant. Then the time dependence of the entropy reflects the information on the nucleus alone, and the correlation is absent. A decomposition of the entropy into contributions from the participating electronic states reveals insight into the state-specific population and nuclear wave packet localization.