  We address extended systems interacting with classical fluctuating
environments and analyze the use of quantum probes to discriminate local noise,
described by independent fluctuating fields, from common noise, corresponding
to the interaction with a common one. In particular, we consider a bipartite
system made of two non interacting harmonic oscillators and assess
discrimination strategies based on homodyne detection, comparing their
performances with the ultimate bounds on the error probabilities of
quantum-limited measurements. We analyze in details the use of Gaussian probes,
with emphasis on experimentally friendly signals. Our results show that a joint
measurement of the position-quadrature on the two oscillators outperforms any
other homodyne-based scheme for any input Gaussian state.
