  Given a finite number $N$ of copies of a qubit state we compute the maximum
fidelity that can be attained using joint-measurement protocols for estimating
its purity. We prove that in the asymptotic $N\to\infty$ limit,
separable-measurement protocols can be as efficient as the optimal
joint-measurement one if classical communication is used. This in turn shows
that the optimal estimation of the entanglement of a two-qubit state can also
be achieved asymptotically with fully separable measurements. The relationship
between our global Bayesian approach and the quantum Cramer-Rao bound is also
discussed.
