Nanoscale structural heterogeneities were recently revealed in computational and experimental studies of calcium silicate hydrates in hardened cement pastes. In this work their consequences for the mechanics are analyzed by computing local pressures in model samples at different overall densities, corresponding to different initial water-to-cement ratios. The correlations between pore size distributions, local density, local cohesive energy, and local pressure clearly show how in these materials structural heterogeneities may be the origin of significant mechanical heterogeneities. The results indicate that even at high density pressure, heterogeneities develop during the densification of cement hydrates and result in the coexistence of regions of high positive pressure with regions of negative pressure, in spite of the overall mechanical stability of the samples. Furthermore, the regions of negative pressure, prone to mechanical instabilities and local plastic processes, tend to be localized close to the surface of large mesopores and hence to be more significant at higher initial water content.