In this paper, we discuss a framework for the distributed compression of vector sources, based on our previous work on distributed transform coding. In particular, our goal is to develop a strategy of first applying a suitable distributed Karhunen-Loeve transform, whereafter each component can be andled by standard distributed compression techniques. In the present paper, we first study the scenario where all but one terminal furnish a noisy approximation of their observation. For the case where the underlying vector is Gaussian, and the added noise is also Gaussian, we establish that indeed, it is optimal for the last terminal to apply a (local) transform to its observations, and to separately compress each component in the transform domain. Then, we outline how this leads to a general simple distributed compression strategy for Gaussian vector sources: Each terminal applies a suitable local transform to its observations, and encodes the resulting components separately in a Wyner-Ziv fashion, i.e., treating the compressed descriptions of all other terminals as side information available to the decoder. This achieves the best known performance. The optimum performance in unknown to date.