To evade the well-known impossibility of unconditionally secure quantum two-party computations, previous quantum private comparison protocols have to adopt a third party. Here, we study how far we can go with two parties only. We propose a very feasible and efficient protocol. Intriguingly, although the average amount of information leaked cannot be made arbitrarily small, we find that this average will not exceed 14 bits for any length of the bit-string being compared.