We consider discrete memoryless compound multiple access and interference channels with correlated sources and correlated side information at the receivers, and investigate necessary and sufficient conditions for lossless transmission. We first give sufficient conditions for the most general setting, and then show that these conditions are also necessary for both channels under certain assumptions on the side information and the interference. In particular, we generalize the notion of strong interference to take into account the correlation among the sources and side information. We prove the optimality of 'informational' or 'operational' source-channel separation for certain special cases. While informational separation results in independent source and channel encoding and decoding; operational separation corresponds to separation at the encoder, while decoding is done jointly. To our knowledge, these results constitute the first source-channel separation results for interference and compound multiple access channels with correlated sources and side information.