This paper addresses the problem of computing the intersection of regular languages in a privacy-preserving fashion. Private set intersection has been addressed earlier in the literature, but for finite sets only. We discuss the various possibilities for solving the problem efficiently, and argue for an approach based on minimal deterministic finite automata (DFA) as a suitable, non-leaking representation of regular language intersection. We propose two different algorithms for DFA minimization in a secure multiparty computation setting, illustrating different aspects of programming based on universal composability and the constraints this sets on existing algorithms. The implementation of our algorithms is based on the programming language SECREC, executing on the SHAREMIND platform for secure multiparty computation. As one application domain we consider fusion of virtual enterprise business processes.