  We aim to interpret the radio emission of stellar bow shocks and assess under
which conditions it could be either thermal or non-thermal, and how to use the
observational data to infer their physical properties. We used an extended
non-thermal emission model for stellar bow shocks for which we incorporated a
consistent calculation of the thermal emission from the forward shock. We
fitted this model to the available radio data, including largely unexplored
data at low frequencies. In addition, we used a simplified one-zone model to
estimate the gamma-ray emission from particles escaping the bow shocks. We can
only explain the radio data from (BD+43{\deg}3654 and BD+60{\deg}2522) assuming
a hard electron energy distribution below $\sim$1 GeV, a high efficiency of
conversion of (shocked) wind kinetic power into relativistic electrons, and a
relatively high magnetic-to-thermal pressure ratio. In the other systems, the
interpretation of the observed flux density is ambiguous, although a
non-thermal scenario is also favoured. We also show how complementary
observations at other frequencies can allow us to place stronger constraints in
the model. We also estimated the gamma-ray fluxes from the HII regions around
the bow shocks of BD+43{\deg}3654 and BD+60{\deg}2522, and obtained
luminosities at GeV energies of $\sim 10^{33}$~erg\,s$^{-1}$ and
$10^{32}$~erg\,s$^{-1}$, respectively. Stellar bow shocks can potentially be
very efficient particle accelerators. This work provides multi-wavelength
predictions of their emission and demonstrates the key role of low-frequency
radio observations in unveiling particle acceleration processes. The prospects
of detections with next-generation observatories such as SKA and ngVLA are
promising. Finally, BD+43{\deg}3654 may be detected in GeV in the near future,
while bow shocks in general may turn out to be non-negligible sources of
low-energy cosmic rays.
