  We study theories of gravity including, in addition to the metric, several
scalar fields in the gravitational sector. The particularity of this work is
that we allow for direct couplings between these gravitating scalars and the
matter sector, which can generally be different for the source and the probe of
gravity, in addition to the universal interactions generated by the Jordan
frame metric. The weak gravity regime of this theory, which would describe
solar-system experiments, is studied using the parametrized post-Newtonian
(PPN) formalism. We derive the expression of the ten parameters of this
formalism. Among them, $\zeta_3$ and $\zeta_4$ are modified with respect to
their values in the theories without direct couplings. This fact holds even
after eliminating the direct couplings between the gravitating scalars and the
energy density of the source, by redefinition of the Jordan frame. All other
PPN parameters are insensitive to the direct couplings once in the correctly
identified Jordan frame. When direct couplings are different for the source and
the probe of gravity, they make non-relativistic probes deviate from the
geodesics of the PPN metric in this frame, already at Newtonian order. Such
couplings would thus be directly detectable and would have excluded by
experiments. This shows that, contrary to the claims in the recent literature,
it is impossible to screen the presence of gravitating scalars relying only on
a curved target space and direct couplings to matter.
