The decay of silver and hafnium atoms with a vacancy in the K shell by the emission of photon pairs which continuously share the transition energy was studied. Radioactive decay of 109Cd and of 179Ta was used to generate K-shell vacancy states in silver and hafnium atoms, respectively. A pair of germanium detectors in a 180° geometry and a fast-slow coincidence system with a (128×512×512)-channel three-parameter pulse-height analyzer were used in the measurements. Two-dimensional spectra of numbers of events as functions of amplitudes of coincident pulses from the two detectors were analyzed. Surface fitting of two-dimensional spectra and curve fitting of sum spectra were applied to deduce the number of events. The differential transition probabilities of 2s→1s, 3s→1s, 3d→1s, and 4sd→1s two-photon decay per decay of a K-shell vacancy have been determined. For silver they are compared with the results of relativistic self-consistent-field calculations of Mu and Crasemann [Phys. Rev. A 38, 4585 (1988)] and of Tong, Li, Kissel, and Pratt [Phys. Rev. A 42, 1442 (1990)]. The results for silver and hafnium are compared with the results of nonrelativistic calculations for hydrogenic silver and hafnium ions. The 3d→1s two-photon decays show the resonance effect, in accordance with the theoretical predictions.