Business process matching is the activity of checking whether a given business process can interoperate with another one in a correct manner. In case the check fails, it is desirable to obtain information about how the first process can be corrected with as few modifications as possible to achieve interoperability. In case the two business processes belong to two separate enterprises that want to build a virtual enterprise, business process matching based on revealing the business processes poses a clear threat to privacy, as it may expose sensitive information about the inner operation of the enterprises. In this paper we propose a solution to this problem for business processes described by means of service automata. We propose a measure for similarity between service automata and use this measure to devise an algorithm that constructs the most similar automaton to the first one that can interoperate with the second one. To achieve privacy, we implement this algorithm in the programming language SecreC, executing on the Sharemind platform for secure multiparty computation. As a result, only the correction information is leaked to the first enterprise and no more.