This paper addresses lossy transmission of a common source over a broadcast channel when there is correlated side information at the receivers. The quadratic Gaussian and binary Hamming cases are especially targeted. Using ideas from the lossless version of the problem, i.e., Slepian-Wolf coding over broadcast channels, and dirty paper coding, several digital schemes are proposed and their single-letter distortion tradeoffs are characterized. These schemes use layered coding where the common layer information is intended for both receivers and the refinement information is destined only for the receiver that is chosen using an appropriately defined "combined" channel/side information quality measure. When this quality is constant at each receiver, all the new schemes converge and become optimal. When the source and the channel bandwidths are equal, it is shown that one of the proposed schemes outperforms all the others as well as separate coding. For the quadratic Gaussian p...