It has been observed that mutualistic bipartite networks have a nested structure of interactions. In addition, the degree distributions associated with the two guilds involved in such networks (e.g., plants and pollinators or plants and seed dispersers) approximately follow a truncated power law (TPL). We show that nestedness and TPL distributions are intimately linked, and that any biological reasons for such truncation are superimposed to finite size effects. We further explore the internal organization of bipartite networks by developing a self-organizing network model (SNM) that reproduces empirical observations of pollination systems of widely different sizes. Since the only inputs to the SNM are numbers of plant and animal species, and their interactions (i.e., no data on local abundance of the interacting species are needed), we suggest that the well-known association between species frequency of interaction and species degree is a consequence rather than a cause, of the observed network structure.