  Models of modified gravity introduce extra degrees of freedom, which for
consistency with the data, should be suppressed at observable scales. In the
models that share properties of massive gravity such a suppression is due to
nonlinear interactions: An isolated massive astrophysical object creates a halo
of a nonzero curvature around it, shielding its vicinity from the influence of
the extra degrees of freedom. We emphasize that the very same halo leads to a
screening of the gravitational mass of the object, as seen by an observer
beyond the halo. We discuss the case when the screening could be very
significant and may rule out, or render the models observationally interesting.
