We investigate local fluctuations in the out of equilibrium dynamics
of a simple glass former, a binary Lennard-Jones system, after a
quench to the glass state. Motivated by recent experiments that probe
fluctuations in glassy materials at distance scales of the order of
tens of nanometers, we define the local (i.e. averaged over a
small region in the sample) two-time mean square displacement
Δr(t,tw) and two-time intermediate scattering function Cr(t,tw). We study the probability distributions for the values of these local two-time functions as a function of both time variables. We discuss the connection between the scaling behavior of the probability distributions and a soft mode controlling the slow dynamics.