  On 2019 April 25.346 and 26.640 UT the LIGO and Virgo gravitational wave (GW)
observatories announced the detection of the first candidate events in
Observing Run 3 that contain at least one neutron star. S190425z is a likely
binary neutron star (BNS) merger at $d_L = 156 \pm 41$ Mpc, while S190426c is
possibly the first NS-BH merger ever detected, at $d_L = 377 \pm 100$ 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/NIR kilonova emission from the BNS merger GW170817, to
theoretical kilonova models, and to short GRB 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 short GRBs, 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 $\gamma$-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 $\gamma$-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 deg$^2$ are likely essential.
