We study the problems of reaching a specific control state, or converging to a set of target states, in networks with a parameterized number of identical processes communicating via broadcast. To reflect the distributed aspect of such networks, we restrict our attention to executions in which all the processes must follow the same local strategy that, given their past performed actions and received messages, provides the next action to be performed. We show that the reachability and target problems under such local strategies are NP-complete, assuming that the set of receivers is chosen non-deterministically at each step. On the other hand, these problems become undecidable when the communication topology is a clique. However, decid-ability can be regained for reachability under the additional assumption that all processes are bound to receive the broadcast messages.