The existence of states enjoying a weak form of the Reeh-Schlieder property has been recently established on curved backgrounds and in the framework of locally covariant quantum field theory. Since only the example of a real scalar field has been discussed, we extend the analysis to the case of massive and massless free fields either of spin 1/2 or of spin 1. In the process, it is also shown that both the vector potential and the Proca field can be described as a locally covariant quantum field theory.