Suppose $A$, $B$ are finite sets of positive integers.
Let $$\mathcal{S}_n = \{C \subset [1,n] \, : \, A+C = B+C \}, $$ and denote $a_n = |\mathcal{S}_n|$.
Note that for any $X \in \mathcal{S}_n$ and $Y \in \mathcal{S}_m$, the set $X \cup (Y+n) \subset [1,m+n]$ is in $\mathcal{S}_{m+n}$, hence $a_m \cdot a_n \le a_{m+n}$ for all $m$, $n$.
By Fekete's Lemma, it follows that $$m(A, B) := \lim_{n\to\infty} \sqrt[n]{a_n}$$ exists and is finite, in particular $0\le m(A, B) \le 2$.
If $A$ and $B$ do not have the same least and greatest elements, then $m(A,B) = 0$; if they do, then $m(A,B) \ge 1$. If $A = B$, then $m(A,B) = 2$.
What can we say about $m(A,B)$ in general? Can we compute it given $A$ and $B$?