We study the unintentional H impurities in relation to the free electron properties of state-of-the-art InN films grown by molecular beam epitaxy (MBE). Enhanced concentrations of H are revealed in the near surface regions of the films, indicating postgrowth surface contamination by H. The near surface hydrogen could not be removed upon thermal annealing and may have significant implications for the surface and bulk free electron properties of InN. The bulk free electron concentrations were found to scale with the bulk H concentrations while no distinct correlation with dislocation density could be inferred, indicating a major role of hydrogen for the unintentional conductivity in MBE InN.