  Various observations are revealing the widespread occurrence of fast and
powerful winds in active galactic nuclei (AGNs) that are distinct from
relativistic jets, likely launched from accretion disks and interacting
strongly with the gas of their host galaxies. During the interaction, strong
shocks are expected to form that can accelerate non-thermal particles to high
energies. Such winds have been suggested to be responsible for a large fraction
of the observed extragalactic gamma-ray background (EGB) in the GeV-TeV range
and the diffuse neutrino background in the PeV range, via the decay of neutral
and charged pions generated in inelastic $pp$ collisions between protons
accelerated by the forward shock and the ambient gas. However, previous studies
did not properly account for processes such as adiabatic losses that may reduce
the gamma-ray and neutrino fluxes significantly. We evaluate the production of
gamma-rays and neutrinos by AGN-driven winds in some detail by modeling their
hydrodynamic and thermal evolution, including the effects of their
two-temperature structure. We find that they can only account for less than
$\sim 30$% of the EGB flux, as otherwise the model would violate the
independent upper limit derived from the diffuse isotropic gamma-ray
background. If the neutrino spectral index is steep with $\Gamma\gtrsim 2.2$, a
severe tension with the isotropic gamma-ray background would arise as long as
the winds contribute more than $20$% of the IceCube neutrino flux in the
$10-100$TeV range. Nevertheless, at energies $\gtrsim100$~TeV, we find that the
IceCube neutrino flux may still be accountable by AGN-driven winds if the
spectral index is as small as $\Gamma\sim2.0-2.1$. The detectability of
gamma-ray point sources also provides important constraints on such scenarios.
