Note that intersections using HLLs don't work in general. The error rates become gigantic for sets that have small intersections, or whose cardinality differs greatly.

A general solution to cardinality estimations for intersections is to just use HLLs for union operations, and keep a MinHash sketch for each key as well to perform intersections.

There is a very good analysis of this technique over at http://tech.adroll.com/blog/data/2013/07/10/hll-minhash.html