We study preferential attachment models where vertices enter the network with i.i.d. random numbers of edges that we call the out-degree. We identify the local limit of such models, substantially extending the work of Berger et al.(2014). The degree distribution of this limiting random graph, which we call the random Pólya point tree, has a surprising size-biasing phenomenon. Many of the existing preferential attachment models can be viewed as special cases of our preferential attachment model with i.i.d. out-degrees. Additionally, our models incorporate negative values of the preferential attachment fitness parameter, which allows us to consider preferential attachment models with infinite-variance degrees. Our proof of local convergence consists of two main steps: a Pólya urn description of our graphs, and an explicit identification of the neighbourhoods in them. We provide a novel and explicit proof to establish a coupling between the preferential attachment model and the Pólya urn graph. Our result proves a density convergence result, for fixed ages of vertices in the local limit.