We consider distributed estimation for a geographically dispersed sensor network, where sensors collect observations that are linearly pre-processed and transmitted over dimensionality-constrained channels. A central processor utilizes the received sensor data to obtain a linear estimate of the desired signal. In this scenario, we consider the optimal preprocessing at the sensors under a mean squared error (MSE) metric. In the single-sensor case, applying a modification of Sakrison's separation principle we show that the optimal preprocessing can be decomposed into two steps: a LMMSE estimate followed by a (linear) MSE optimal dimensionality reduction of the estimate. The latter is readily obtained as the well-known Karhunen-Loeve transform (KLT). Under the multi-sensor scenario, we extend this result to show that given the pre-processing at other nodes, each node's optimal linear pre-processing again reduces to a side-informed linear estimation followed by a side-informed version of the KLT. The separation perspective thus provides a simple and intuitive derivation of the optimal linear pre-processing under reduced dimensionality channels.