In many applications, there is a need for digital representations of various kinds of content. Such content includes, for example, video, audio, imagery, as well as various kinds of sensor data. To develop compact representations, one needs to take into account the semantics of the content. In particular, the goal is not to enable an exact reconstruction of the content, but rather one that is semantically indistinguishable from original for the applications of interest. In Shannon's formulation of the problem, the semantics are captured through a distortion measure. When such semantic information is shared between encoder and decoder, the fundamental rate-distortion tradeoffs are well understood. This work explores the corresponding tradeoffs when such semantic information is not universally available. We show, for example, that when only the decoder has access to the full semantics, it provides no benefit and may as well be ignored. By contrast, when only the encoder has access to the full semantics, in many cases that is sufficient to do as well as if the decoder had it too. Moreover, we show that systems in which such semantic information is measured at the encoder and shared with the decoder through a side channel -- which is the basis of many perceptual coding systems, for example -- can be particularly inefficient. As an efficient alternative, we introduce low-complexity lattice codes in which there is a fixed codebook but a variable partition to exploit encoder-only side information.