  We study the multiparameter sensitivity bounds of a sensor made by an array
of $d$ spatially-distributed Mach-Zehnder interferometers (MZIs). A generic
single non-classical state is mixed with $d-1$ vacuums to create a $d$-modes
entangled state, each mode entering one input port of a MZI, while a coherent
state enters its second port. We show that local measurements, independently
performed on each MZI, are sufficient to provide a sensitivity saturating the
quantum Cram\'er-Rao bound. The sensor can overcome the shot noise limit for
the estimation of arbitrary linear combinations of the $d$ phase shifts,
provided that the non-classical probe state has an anti-squeezed quadrature
variance. We compare the sensitivity bounds of this sensor with that achievable
with $d$ independent MZIs, each probed with a nonclassical state and a coherent
state. We find that the $d$ independent interferometers can achieve the same
sensitivity of the entangled protocol but at the cost of using additional $d$
non-classical states rather than a single one. When using in the two protocols
the same average number of particles per shot $\bar{n}_T$, we find analytically
a sensitivity scaling $1/\bar{n}_T^2$ for the entangled case which provides a
gain factor $d$ with respect to the separable case where the sensitivity scales
as $d/\bar{n}_T^2$. We have numerical evidences that the gain factor $d$ is
also obtained when fixing the total average number of particles, namely when
optimizing with respect to the number of repeated measurements.
