What is a good distance measure for matching SIFT descriptors depending on the distribution of their noise?
For obtaining distributions or modes in your, a basic approach is to cluster the data. That will of course bias the distribution towards the metric, or in other words the distribution will be tied to the metric. For example, it is possible to quantize the SIFT descriptors by using vocabulary trees to visual words and then plot the 1D histogram of the individual words. That's only one possible way to do this from a large collection of SIFT descriptors. Another approach would use nonparametric methods (Parzen Window, Meanshift...). In either case the distribution will be data-dependent and quite complex.

I think this question also deserves to mention the Root-Sift paper, where authors propose to use the Hellinger distance instead of Euclidean. The implementation is very easy (just take a sqrt) and the results are consistently improved. K-means can still be used. Apparently, Hellinger is a better metric than Euclidean for SIFT.