We show using molecular dynamics simulations that simple diatomic molecules in the glassy state exhibit only limited participation in the Johari-Goldstein (JG) relaxation process. That is, with sufficient cooling local reorientations are essentially frozen for some molecules, while others continue to change their orientation significantly. Thus, the "islands of mobility" concept is valid for these molecular glass-formers deep in the glassy state; only near the glass transition temperature does every molecule undergo the JG process. In contrast, for a linear polymer this dichotomy in the distribution of JG relaxation strengths is absent if any segments are changing their local orientation, all segments are.