In spite of the importance of identification mechanisms in ontology engineering, the Description Logics at the basis of current reasoners do not include modeling features for expressing identification constraints. In this paper, we consider a powerful class of identification constraints, which allow for using roles, inverses, and paths, thus capturing sophisticated forms of identifications often needed in real-world applications. We show that, when used with no limitations, such path-based identification constraints are problematic with respect to effectiveness/efficiency of reasoning. We then propose a restricted form of these constraints, called local, requiring that at least one of the component paths of the concept identifier is a direct property of the concept. We argue that such a restriction is not a severe limitation in practice, and we show that local path-based identification constraints do not increase the complexity of reasoning both in very expressive Description Logics and in the tractable DL-Lite family.