  Let $A_1$ and $A_2$ be randomly chosen subsets of the first $n$ integers of
cardinalities $s_2\geq s_1 = \Omega(s_2)$, such that their sumset $A_1+A_2$ has
size $m$. We show that asymptotically almost surely $A_1$ and $A_2$ are almost
fully contained in arithmetic progressions $P_1$ and $P_2$ with the same common
difference and cardinalities approximately $s_i m/(s_1+s_2)$. We also prove a
counting theorem for such pairs of sets in arbitrary abelian groups. The
results hold for $s_i = \omega(\log^3 n)$ and $s_1+s_2 \leq m = o(s_2/\log^3
n)$. Our main tool is an asymmetric version of the method of hypergraph
containers which was recently used by Campos to prove similar results in the
special case $A=B$.
