Consider a bandwidth constrained sensor network in which a set of distributed sensors and a fusion center (FC) collaborate to estimate an unknown vector. Due to power and cost limitations, each sensor must compress its data in order to minimize the amount of information that need to be communicated to the FC. In this paper, we consider the design of a linear decentralized estimation scheme (DES) whereby each sensor transmits over a noisy channel to the FC a fixed number of real-valued messages which are linear functions of its observations, while the FC linearly combines the received messages to estimate the unknown parameter vector. Assuming each sensor collects data according to a local linear model, we propose to design optimal linear message functions and linear fusion function according to the minimum mean squared error (MMSE) criterion. We show that the resulting design problem is nonconvex and NP-hard in general, and identify two special cases for which the optimal linear DES design problem can be efficiently solved either in closed form or by Semi-definite programming (SDP).