  The dynamics of two interacting quantum particles on a weakly disordered
chain is investigated. Spatial quantum interference between them is
characterized through the statistics of two-particle transition amplitudes,
related to Hanbury Brown-Twiss correlations in optics. The fluctuation profile
of the signal can discern whether the interacting parties are behaving like
identical bosons, fermions, or distinguishable particles. An analog fully
developed speckle regime displaying Rayleigh statistics is achieved for
interacting bosons. Deviations toward long-tailed distributions echo quantum
correlations akin to non-interacting identical particles. In the limit of
strong interaction, two-particle bound states obey generalized Rician
distributions.
