  The general problem of tracer diffusion in non-equilibrium baths is important
in a wide range of systems, from the cellular level to geographical
lengthscales. In this paper, we revisit the archetypical example of such a
system: a collection of small passive particles immersed in a dilute suspension
of non-interacting dipolar microswimmers, representing bacteria or algae. In
particular, we consider the interplay between thermal (Brownian) diffusion and
hydrodynamic (active) diffusion due to the persistent advection of tracers by
microswimmer flow fields. Previously, it has been argued that even a moderate
amount of Brownian diffusion is sufficient to significantly reduce the
persistence time of tracer advection, leading to a significantly reduced value
of the effective active diffusion coefficient $D_A$ compared to the
non-Brownian case. Here, we show by large-scale simulations and kinetic theory
that this effect is in fact only practically relevant for microswimmers that
effectively remain stationary while still stirring up the surrounding fluid,
so-called \emph{shakers}. In contrast, for moderate and high values of the
swimming speed $v_s$, relevant for biological microswimmer suspensions, the
effect of Brownian motion on $D_A$ is negligible, leading to the effects of
advection by microswimmers and Brownian motion being additive. This conclusion
contrasts with previous results from the literature, and encourages a
reinterpretation of recent experimental measurements of $D_A$ for tracer
particles of varying size in bacterial suspensions.
