The principle of local distinguishability states that an arbitrary physical state of a bipartite system can be determined by the combined statistics of local measurements performed on the subsystems. A necessary and sufficient requirement for the local measurements is that each one must be able to distinguish between all pairs of states of the respective subsystems. We show that, if the task is changed into the determination of an arbitrary bipartite pure state, then at least in certain cases it is possible to restrict to local measurements which can distinguish all pure states but not all states. Moreover, we show that, if the local measurements are such that the purity of the bipartite state can be verified from the statistics without any prior assumption, then in these special cases also this property is carried over to the composite measurement. These surprising facts give evidence that the principle of local distinguishability may be expanded beyond its usual applicability.