We report first-principles calculations of local heating in nanoscale junctions formed by a single molecule and a gold point contact. Due to the lower current density and larger heat dissipation, the single molecule heats less than the gold point contact. We also find that, at zero temperature, threshold biases, Vonset, of about 6 mV and 11 mV for the molecule and the point contact, respectively, are required to excite the smallest vibrational mode and generate heat. The latter estimate is in very good agreement with recent experimental results on the same system. At a given external bias V below Vonset, heating becomes noticeable when the background temperature is on the order of approximately e(Vonset - V)/kB. Above Vonset, local heating increases dramatically with increasing bias, mainly due to excitation of longitudinal modes, but is also considerably suppressed by thermal dissipation into the electrodes, provided good thermal contacts exist between the nanostructure and the bulk electrodes.