  Among the most eagerly anticipated opportunities made possible by Advanced
LIGO/Virgo are multimessenger observations of compact mergers. Optical
counterparts may be short-lived so rapid characterization of gravitational wave
(GW) events is paramount for discovering electromagnetic signatures. One way to
meet the demand for rapid GW parameter estimation is to trade off accuracy for
speed, using waveform models with simplified treatment of the compact objects'
spin. We report on the systematic errors in GW parameter estimation suffered
when using different spin approximations to recover generic signals. Component
mass measurements can be biased by $>5\sigma$ using simple-precession waveforms
and in excess of $20\sigma$ when non-spinning templates are employed. This
suggests that electromagnetic observing campaigns should not take a strict
approach to selecting which LIGO/Virgo candidates warrant follow-up
observations based on low-latency mass estimates. For sky localization, we find
searched areas are up to a factor of ${\sim}$2 larger for non-spinning
analyses, and are systematically larger for any of the simplified waveforms
considered in our analysis. Distance biases for the non-precessing waveforms
can be in excess of 100\% and are largest when the spin angular momenta are in
the orbital plane of the binary. We confirm that spin-aligned waveforms should
be used for low-latency parameter estimation at the minimum. Including simple
precession, though more computationally costly, mitigates biases except for
signals with extreme precession effects. Our results shine a spotlight on the
critical need for development of computationally inexpensive precessing
waveforms and/or massively parallel algorithms for parameter estimation.
