The scale of international migration flows depends on moving costs that are, in turn, influenced by host-country policies and by the size of migrant networks at destination. This paper estimates the influence of visa policies and networks upon bilateral migration flows to multiple destinations. We rely on a Poisson pseudo-maximum likelihood estimator to derive estimates that are consistent under more general distributional assumptions on the underlying RUM model than the ones commonly adopted in the literature. We derive bounds for the estimated direct and indirect effects of visa policies and networks that reflect the uncertainty connected to the use of aggregate data, and we show that bilateral migration flows can be highly sensitive to the immigration policies set by other destination countries, an externality that we are able to quantify.