  Numerical models of collisionless shocks robustly predict an electron
distribution comprised of both thermal and non-thermal electrons. Here, we
explore in detail the effect of thermal electrons on the emergent synchrotron
emission from sub-relativistic shocks. We present a complete `thermal +
non-thermal' synchrotron model and derive properties of the resulting spectrum
and light-curves. Using these results we delineate the relative importance of
thermal and non-thermal electrons for sub-relativistic shock-powered
synchrotron transients. We find that thermal electrons are naturally expected
to contribute significantly to the peak emission if the shock velocity is
$\gtrsim 0.2c$, but would be mostly undetectable in non-relativistic shocks.
This helps explain the dichotomy between typical radio supernovae and the
emerging class of `AT2018cow-like' events. The signpost of thermal electron
synchrotron emission is a steep optically-thin spectral index and a $\nu^2$
optically-thick spectrum. These spectral features are also predicted to
correlate with a steep post-peak light-curve decline rate, broadly consistent
with observed AT2018cow-like events. We expect that thermal electrons may be
observable in other contexts where mildly-relativistic shocks are present, and
briefly estimate this effect for gamma-ray burst afterglows and binary neutron
star mergers. Our model can be used to fit spectra and light-curves of events
and accounts for both thermal and non-thermal electron populations with no
additional physical degrees of freedom.
