The production of $\ensuremath{\omega}$ mesons in proton-proton collisions for proton incident energies up to $2.2 \mathrm{GeV}$ is investigated within a meson-exchange model of hadronic interactions. We find a large cancellation between the dominant $\ensuremath{\pi}\ensuremath{\rho}\ensuremath{\omega}$ meson-exchange current and nucleonic current contributions. A comparison with preliminary data from SATURNE calls for the inclusion of off-shell form factors at the $\mathrm{NN}\ensuremath{\omega}$ and $\ensuremath{\pi}\ensuremath{\rho}\ensuremath{\omega}$ production vertices. Due to the present lack of knowledge of these form factors, together with the destructive interference mentioned above, the relative magnitude of the nucleonic and meson-exchange current contributions cannot be determined from existing total cross section data. However, it is shown that the angular distribution of the produced $\ensuremath{\omega}$ mesons provides a unique and clear signature of the magnitude of these currents, thus allowing one to disentangle these two basic reaction mechanisms.