Effective communication requires adapting to the idiosyncratic common ground shared with each communicative partner. We study a particularly challenging instantiation of this problem: the popular game Dixit. We formulate a round of Dixit as a multi-agent image reference game where a (trained) speaker model is rewarded for describing a target image such that one (pretrained) listener model can correctly identify it from a pool of distractors, but another listener cannot. To adapt to this setting, the speaker must exploit differences in the common ground it shares with the different listeners. We show that finetuning an attention-based adapter between a CLIP vision encoder and a large language model in this contrastive, multi-agent setting gives rise to context-dependent natural language specialization from rewards only, without direct supervision. In a series of controlled experiments, we show that the speaker can adapt according to the idiosyncratic strengths and weaknesses of various pairs of different listeners. Furthermore, we show zero-shot transfer of the speaker's specialization to unseen real-world data. Our experiments offer a step towards adaptive communication in complex multi-partner settings and highlight the interesting research challenges posed by games like Dixit. We hope that our work will inspire creative new approaches to adapting pretrained models.