We derive and discuss general physical bounds on the electromagnetic scattering and absorption of passive structures. Our theory, based on passivity and power conservation, quantifies the minimum and maximum allowed scattering for an object that absorbs a given level of power. We show that there is a fundamental tradeoff between absorption and overall scattering suppression for each scattering harmonic, providing a tool to quantify the performance of furtive sensors, regardless of the applied principle for scattering suppression. We illustrate these fundamental limitations with examples of light scattering from absorbing plasmonic nanoparticles and loaded dipole antennas, envisioning applications to the design of cloaked sensors and absorbers with maximized absorption efficiency.