I will discuss a one-dimensional system of bosons or fermions with stochastic nearest-neighbor hopping, modeled as Brownian motion and driven out of equilibrium by boundary injection and removal processes. This setting defines the open Quantum Simple Symmetric Exclusion (QSSEP, for fermions) or Inclusion (QSSIP, for bosons) Process. For these noisy quantum systems, I will present results on the fluctuation statistics of density and current. For large system size, the corresponding large deviation functions are shown to converge with those of their classical counterparts, revealing a form of classical typicality emerging from quantum stochastic dynamics at large scales.