The `random intersection graph with communities' models networks with communities, assuming an underlying bipartite structure of groups and individuals. Each group has its own internal structure described by a (small) graph, while groups may overlap. The group memberships are generated by a bipartite configuration model. The model generalizes the classical random intersection graph model that is included as the special case where each community is a complete graph (or clique). The `random intersection graph with communities' is analytically tractable. We prove a phase transition in the size of the largest connected component based on the choice of model parameters. Further, we prove that percolation on our model produces a graph within the same family, and that percolation also undergoes a phase transition. Our proofs rely on the connection to the bipartite configuration model, however, with the arbitrary structure of the groups, it is not completely straightforward to translate results on the group structure into results on the graph. Our related results on the bipartite configuration model are not only instrumental to the study of the random intersection graph with communities, but are also of independent interest, and shed light on interesting differences from the unipartite case.