  The self-averaging behavior of interacting many-body quantum systems has been
mostly studied at equilibrium. The present work addresses what happens out of
equilibrium, as the increase of the strength of onsite disorder takes the
system to the localized phase. We consider two local and two non-local
quantities of great experimental and theoretical interest. In the delocalized
phase, self-averaging depends on the observable and on the time scale, but the
picture simplifies substantially when localization is reached. In the localized
phase, the local observables become self-averaging at all times, while the
non-local quantities are throughout non-self-averaging. These behaviors are
explained and scaling analysis are provided using the $\ell$-bits model and a
toy model.
