In this paper, we investigate a new practical learning scenario, where the data distributed in different sources/clients are typically generated with various modalities. Existing research on learning from multi-source data mostly assume that each client owns the data of all modalities, which may largely limit its practicability. In light of the expensiveness and sparsity of multimodal data, we propose "checkerboard learning" to jointly learn from fragmented multimodal data in distributed clients. Considering the concerns on data privacy, checkerboard learning aims to impute incomplete multimodal data for diverse downstream tasks without accessing the raw data directly. Local clients could miss different modality combinations. Due to the statistical heterogeneity induced by non-i.i.d. data, the imputation is more challenging since the learned dependencies fail to adapt to the imputation of other clients. In this paper, we provide a novel imputation framework to tackle modality combination heterogeneity and statistical heterogeneity simultaneously, called ``collaborative adaptation''. In particular, for two observed modality combinations from two clients, we learn the transformations between their maximal intersection and other modalities by proposing a novel ELBO. We improve the worst-performing required transformations through a Pareto min-max framework. In extensive experiments, we demonstrate the superiority of the proposed method compared to existing related methods on benchmark data sets and a real-world clinical data set.