On 2019 April 25.346 and 26.640 UT the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo gravitational-wave (GW) observatory announced the detection of the first candidate events in Observing Run 3 that contained at least one neutron star (NS). S190425z is a likely binary neutron star (BNS) merger at dL = 156 41 Mpc, while S190426c is possibly the first NS-black hole (BH) merger ever detected, at dL = 377 100 Mpc, although with marginal statistical significance. Here we report our optical follow-up observations for both events using the MMT 6.5 m telescope, as well as our spectroscopic follow-up of candidate counterparts (which turned out to be unrelated) with the 4.1 m SOAR telescope. We compare to publicly reported searches, explore the overall areal coverage and depth, and evaluate those in relation to the optical/near-infrared (NIR) kilonova emission from the BNS merger GW170817, to theoretical kilonova models, and to short gamma-ray burst (SGRB) afterglows. We find that for a GW170817-like kilonova, the partial volume covered spans up to about 40% for S190425z and 60% for S190426c. For an on-axis jet typical of SGRBs, the search effective volume is larger, but such a configuration is expected in at most a few percent of mergers. We further find that wide-field γ-ray and X-ray limits rule out luminous on-axis SGRBs, for a large fraction of the localization regions, although these searches are not sufficiently deep in the context of the γ-ray emission from GW170817 or off-axis SGRB afterglows. The results indicate that some optical follow-up searches are sufficiently deep for counterpart identification to about 300 Mpc, but that localizations better than 1000 deg2 are likely essential.