  Each agent in a network makes a local observation that is linearly related to
a set of public and private parameters. The agents send their observations to a
fusion center to allow it to estimate the public parameters. To prevent leakage
of the private parameters, each agent first sanitizes its local observation
using a local privacy mechanism before transmitting it to the fusion center. We
investigate the utility-privacy tradeoff in terms of the Cram\'er-Rao lower
bounds for estimating the public and private parameters. We study the class of
privacy mechanisms given by linear compression and noise perturbation, and
derive necessary and sufficient conditions for achieving arbitrarily strong
utility-privacy tradeoff in a multi-agent system for both the cases where prior
information is available and unavailable, respectively. We also provide a
method to find the maximum estimation privacy achievable without compromising
the utility and propose an alternating algorithm to optimize the
utility-privacy tradeoff in the case where arbitrarily strong utility-privacy
tradeoff is not achievable.
