We consider multiterminal source coding with a single encoder and multiple decoders where either the encoder or the decoders can take cost-constrained actions which affect the quality of the side information present at the decoders. For the scenario where decoders take actions, we characterize the rate-cost tradeoff region for lossless source coding, and give an achievability scheme for lossy source coding for two decoders which is optimum for a variety of special cases of interest. For the case where the encoder takes actions, we characterize the rate-cost tradeoff for a class of lossless source coding scenarios with multiple decoders. Finally, we also consider extensions to other multiterminal source coding settings with actions, and characterize the rate-distortion-cost tradeoff for a case of successive refinement with actions.