We propose a random bipartite graph with weights assigned to both parts of the vertex sets. Edges are formed independently with probabilities that depend on these weights. This bipartite graph naturally gives rise to a random intersection graph which has nontrivial clustering properties and inhomogeneous vertex degrees. We focus on the situation where the weights are themselves i.i.d. random variables. In the so-called moderate clustering regime, we identify three types of scaling limit for the large connected components in the graphs at criticality, depending on the tail behaviours of the weight distributions of both parts.