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We consider the allocation of indivisible objects when agents have preferences over their own allocations, but share the ownership of the resources to be distributed. Examples might include seats in public schools, faculty offices, and time slots in public tennis courts. Given an allocation, groups of agents who would prefer an alternative allocation might challenge it. An assignment is popular if it is not challenged by another one. By assuming that agents' ability to challenge allocations can be represented by weighted votes, we characterize the conditions under which popular allocations might exist and when these can be implemented via strategy-proof mechanisms. Serial dictatorships that use orderings consistent with the agents' weights are not only strategy-proof and Pareto efficient, but also popular, whenever these assignments exist. We also provide a new characterization for serial dictatorships as the only mechanisms that are popular, strategy-proof, non-wasteful, and satisfy a consistency condition.