Two remotely located agents, Alice and Bob, observe an unlimited number of i.i.d. samples, each of a different part of a Gaussian vector. Alice can send a fixed number of bits on average to Bob, who in turn wants to estimate the correlations between the two parts of the vector. In the case where the agents observe scalar Gaussian random variables with unknown correlation, we obtain two constructive and simple unbiased estimators whose performance coincides with a known but nonconstructive random coding result of Zhang and Berger. In the vector case, which was not treated before, we obtain a nontrivial multidimensional extension that employs the coupling between the correlations to yield better performance. We also discuss application of our technique to cases where the underlying distribution is not fully known.