Two-proton emissions from the excited states of Ne-17,Ne-18, P-28 and S-28,S-29 were investigated experimentally by the radioactive beams bombarding on the Au-197 target via the Coulomb excitation. The complete-kinematics measurements were actualized by the stacks of silicon-strip detectors and CsI+PIN array detectors. The invariant mass of final systems as well as the relative momentum, opening angle, and relative energy of the two emitted protons was reconstructed under the framework of relativistic kinematics. Visible proton-proton correlations were observed in these systems. The mechanisms of two-proton emission were analyzed in a simple schematic model, in which the extreme decay modes like He-2 cluster emission, three-body phase-space decay, and two-body sequential decay were taken into account. With the help of Monte-Carlo simulations, the two protons emitted from the 6.15 MeV excited state of Ne-18 and the excited states in the energy region of 9.6-10.4 MeV of S-29, respectively, exhibited prominent features of He-2 cluster decay while for the other cases, no obvious diproton emissions were observed.