In this paper we investigate biometric person identification. We model this process of person identification as multiple hypothesis testing and consider performance measures that can be attained in such a protocol in terms of exponents of average error probability. A special attention is paid to the privacy related issues. In particular, we consider performance/privacy trade-off due to the indirect identification that is based on data projection of an original data to a secret subspace. Finally, we approximate the obtained performance limits using properties of random projections. Finally, experimental simulations are used to exemplify our findings.