We show numerically that the response of simple amorphous solids (elastic networks and particle packings) to a local force dipole is characterized by a lengthscale c that diverges as unjamming is approached as c ∼ (z - 2d)-1/2, where z ≥ 2d is the mean coordination, and d is the spatial dimension, at odds with previous numerical claims. We also show how the magnitude of the lengthscale c is amplified by the presence of internal stresses in the disordered solid. Our data suggests a divergence of c ∼ (pc - p)-1/4 with proximity to a critical internal stress pc at which soft elastic modes become unstable. This journal is © the Partner Organisations 2014.