Source coding with a side information "vending machine" is a recently proposed framework in which the statistical relationship between the side information available at the decoder and the source sequence can be controlled by the decoder based on the message received from the encoder. In this paper, the characterization of the optimal rate-distortion performance as a function of the cost associated with the control actions is extended from the previously studied point-to-point set-up to two multiterminal models. First, a distributed source coding model is studied, in which two encoders communicate over rate-limited links to a decoder, whose side information can be controlled based on the control actions selected by one of the encoders. The rate-distortion-cost region is characterized under the assumption of lossless reconstruction of the source encoded by the node that does not control the side information. Then, a three-node cascade scenario is investigated, in which the last node has controllable side information. The rate-distortion-cost region is derived for general distortion requirements and under the assumption of "causal" availability of side information at the last node.