To analyze the Johari-Goldstein (JG) secondary process, dielectric relaxation measurements were carried out on poly(methyl methacrylate) (PMMA) and poly(thiomethyl methacrylate) (PTMA). The latter has a sulfur atom replacing the oxygen in the ester of PMMA, making the pendant group much less polar. This weakens the intensity of the secondary relaxation, enabling facile resolution of the JG and primary structural relaxation peaks. We find that the JG relaxation time of PTMA has a substantially larger activation energy (factor of 2) in the liquid compared to the glassy state. Although deconvolution of the peaks in PMMA is more ambiguous due to the large dielectric strength of its JG relaxation, applying the same analysis leads to a very similar result. In light of previous results, we conclude very generally from their temperature-dependence that JG secondary motions sense the glass transition, consistent with their role as precursor to structural relaxation.