Mass screening in modified gravity

Models of modified gravity introduce extra degrees of freedom, which for a 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.