Navigational path query, one of the most popular location-based services (LBSs), determines a route from a source to a destination on a road network. However, issuing path queries to some non-trustworthy service providers may pose privacy threats to the users. For instance, given a query requesting for a path from a residential address to a psychiatrist, some adversaries may deduce "who is related to what disease". In this paper, we present an obfuscator framework that reduces the likelihood of path queries being revealed, while supporting different user privacy protection needs and retaining query evaluation efficiency. The framework consists of two major components, namely, an obfuscator and an obfuscated path query processor. The former formulates obfuscated path queries by intermixing true and fake sources and destinations and the latter facilitates efficient evaluation of the obfuscated path queries in an LBS server. The framework supports three types of obfuscated path queries, namely, independent obfuscated path query, shared obfuscated path query, and anti-collusion obfuscated path query. Our proposal strikes a balance between privacy protection strength and query processing overheads, while enhancing privacy protection against collusion attacks. Finally, we validate the proposed ideas and evaluate the performance of our framework based on an extensive set of empirical experiments.