  We develop in this paper general techniques to analyze local combinatorial
structures in product sets of two subsets of a countable group which are
"large" with respect to certain classes of (not necessarily invariant) means on
the group. As applications of our methods, we extend and quantify a series of
recent results by Jin, Bergelson-Furstenberg-Weiss, Beiglb\"ock-Bergelson-Fish,
Griesmer and DiNasso-Lupini to general countable groups.
