Theoretically, bound binaries of massive black holes are expected as the natural outcome of mergers of massive galaxies. From the observational side, however, massive black hole binaries remain elusive. Velocity shifts between narrow and broad emission lines in quasar spectra are considered a promising observational tool to search for spatially unresolved, dynamically bound binaries. In this series of papers we investigate the nature of such candidates through analyses of their spectra, images and multi-wavelength spectral energy distributions. Here we investigate the properties of the optical spectra, including the evolution of the broad line profiles, of all the sources identified in our previous study. We find a diverse phenomenology of broad and narrow line luminosities, widths, shapes, ionization conditions and time variability, which we can broadly ascribe to 4 classes based on the shape of the broad line profiles: 1) Objects with bell-shaped broad lines with big velocity shifts (>1000 km/s) compared to their narrow lines show a variety of broad line widths and luminosities, modest flux variations over a few years, and no significant change in the broad line peak wavelength. 2) Objects with double-peaked broad emission lines tend to show very luminous and broadened lines, and little time variability. 3) Objects with asymmetric broad emission lines show a broad range of broad line luminosities and significant variability of the line profiles. 4) The remaining sources tend to show moderate to low broad line luminosities, and can be ascribed to diverse phenomena. We discuss the implications of our findings in the context of massive black hole binary searches.
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