We study dissipative properties of systems composed of two components one of which is highly lossy and the other is lossless. One of the principal result is that the dissipation causes modal dichotomy, i.e., splitting of the eigenmodes into two distinct classes according to their dissipative properties: high-loss and low-loss modes. Interestingly, larger losses in the lossy component make the entire composite less lossy, the dichotomy more pronounced, low-loss modes less lossy, and high-loss modes less accessible to external excitations. We also have carried out an exhaustive analytical study of the system quality factor. This is joint work with Alexander Figotin.