We report an investigation of the local modes of hydrogen and deuterium atoms by means of inelastic neutron scattering at 10 K and 100 K in some hydride and deuteride monohalides of the rare-earth elements, REXH_x, with RE=Y and Tb, X=Cl and Br and x approximately=0.8. These compounds contain hydrogen/deuterium atoms in distorted tetrahedral metal-atom voids. The hydride spectra exhibit two modes centred at about 125 meV and 90 meV for YBrD_x. For the deuterium compounds YBrD_x (x approximately=0.8) second-harmonic transitions are observed. The energy splitting of the two modes is of the order of 6-10 meV. The low-energy mode is about twice as intense as the high-energy mode. The splitting of the modes is consistent with local C_3v symmetry and analysed in terms of a model suggested recently by Bennington et al. Broadenings of the modes are observed and attributed to a weak splitting of the doubly degenerate low-energy mode due to a reduction of local symmetry. To relate the spectra to crystallographic properties a powder neutron-diffraction experiment and a refinement of the crystal structure of YBrD_0.79 have also been performed.