We have measured the thermal properties of suspended membranes from 10 to 300 K for two amplitudes of internal stress (about 0.1 and 1 GPa) and for two different thicknesses (50 and 100 nm). The use of the original $3\omega$-Volklein method has allowed the extraction of both the specific heat and the thermal conductivity of each SiN membrane over a wide temperature range. The mechanical properties of the same substrates have been measured at helium temperatures using nanomechanical techniques. Our measurements show that the thermal transport in freestanding SiN membranes is not affected by the presence of internal stress. Consistently, mechanical dissipation is also unaffected even though $Q$'s increase with increasing tensile stress. We thus demonstrate that the theory developed by Wu and Yu [J. Wu and C. C. Yu, Phys. Rev. B 84, 174109 (2011)] does not apply to this amorphous material in this stress range. On the other hand, our results can be viewed as a natural consequence of the “dissipation dilution” argument [Y. L. Huang and P. R. Saulson, Rev. Sci. Instrum. 69, 544 (1998)], which has been introduced in the context of mechanical damping.