Let $L \subseteq A^*$ be a language, and define $f_L\colon A^* \times A^* \to \{0, 1\}$ by $f_L(x, y) = 1$ iff $x\cdot y \in L$. I'm searching for a reference for:
Proposition. $L$ is regular iff the deterministic communication complexity of $f_L$ is constant.
In other words, $L$ is regular iff there exists a two-player protocol $P$ for $f_L$ such that the function $$n \mapsto \max\{\text{comm}(P, x, y) : |x\cdot y| = n\}$$ is bounded by a constant, where $\text{comm}(P, x, y)$ is the number of bits exchanged by Alice and Bob when Alice receives $x$ and Bob $y$, following the protocol $P$.
The only place I could find that is in George Hauser's PhD thesis, 1989, available here, where he also generalizes that to other distributions of the input $x\cdot y$ among Alice and Bob, such that the number of "cuts" is constant.