A partial answer is contained in Betti's Formal theory of internal categories (page 49), where he states that the bicategory $\mathbf{Dist}(\mathcal E)$ of $\mathcal E$-internal distributors is the free completion of $\mathbf{Cat}(\mathcal E)$ under right extensions, at least under the assumption that the underlying class of objects is preserved. It seems plausible this result could be generalised, but likely not to arbitrary 2-categories.

A partial answer is contained in Betti's Formal theory of internal categories (page 49), where he states that the bicategory $\mathbf{Dist}(\mathcal E)$ of $\mathcal E$-internal distributors is the free completion of $\mathbf{Cat}(\mathcal E)$ under right extensions, at least under the assumption that the underlying class of objects is preserved. It seems plausible this approach could be generalised, but likely not to arbitrary 2-categories, which would probably require a different approach.