I prefer to think of profunctors $\mathbb{A} \to \mathbb{B}$ (which for me means a functor $\mathbb{A} \to [\mathbb{B}^{\textrm{op}}, \textbf{Set}]$) as secretly being colimit-preserving functors $[\mathbb{A}^\textrm{op}, \textbf{Set}] \to [\mathbb{B}^\textrm{op}, \textbf{Set}]$. Indeed, the presheaf topos $[\mathbb{A}^\textrm{op}, \textbf{Set}]$ is characterised by the following universal property:

- For every cocomplete category $\mathcal{E}$, the functor $\textbf{Cocont}([\mathbb{A}^\textrm{op}, \textbf{Set}], \mathcal{E}) \to [\mathbb{A}, \mathcal{E}]$ sending a colimit-preserving functor $F : [\mathbb{A}^\textrm{op}, \textbf{Set}] \to \mathcal{E}$ to the composite $F Y : \mathbb{A} \to \mathcal{E}$, where $Y : \mathbb{A} \to [\mathbb{A}^\textrm{op}, \textbf{Set}]$ is the Yoneda embedding, is fully faithful and surjective on objects.

In particular, $[\mathbb{B}^\textrm{op}, \textbf{Set}]$ is a cocomplete category, so the category of functors $\mathbb{A} \to [\mathbb{B}^\textrm{op}, \textbf{Set}]$ is pseudonaturally equivalent to the category of colimit-preserving functors $[\mathbb{A}^\textrm{op}, \textbf{Set}] \to [\mathbb{B}^\textrm{op}, \textbf{Set}]$, and one can check that this makes the bicategory of small categories and profunctors 3-equivalent to the 2-category of presheaf toposes and colimit-preserving functors. (One may think of the bicategory of profunctors as being a bicategorical analogue of the Kleisli category for the pseudomonad that sends a small category to its free cocompletion... but such a pseudomonad cannot be defined on the 2-category of small categories, so this is not literally true.)

Thus, a monad on $\mathbb{A}$ in the bicategory of profunctors is the same thing as a monad on $[\mathbb{A}^\textrm{op}, \textbf{Set}]$ whose underlying endofunctor preserves colimits. It is certainly true that every category $\mathbb{B}$ that admits an identity-on-objects functor $\mathbb{A} \to \mathbb{B}$ induces such a monad: indeed, by precomposition, we get a conservative functor $[\mathbb{B}^\textrm{op}, \textbf{Set}] \to [\mathbb{A}^\textrm{op}, \textbf{Set}]$ with a left and right adjoint, and thus by Beck's monadicity theorem, $[\mathbb{B}^\textrm{op}, \textbf{Set}]$ is monadic over $[\mathbb{A}^\textrm{op}, \textbf{Set}]$.

On the other hand, suppose we have a monad $\mathsf{T}$ on $[\mathbb{A}^\textrm{op}, \textbf{Set}]$ whose underlying endofunctor preserves colimits. Let $\mathcal{B}$ be the category of $\mathsf{T}$-algebras. By standard nonsense, the forgetful functor $U : \mathcal{B} \to [\mathbb{A}^\textrm{op}, \textbf{Set}]$ is conservative, has a left adjoint $F : [\mathbb{A}^\textrm{op}, \textbf{Set}]$, and creates both limits and colimits. Using Weber's nerve theorem, it follows that the full subcategory $\mathbb{B}$ of $\mathcal{B}$ spanned by the $\mathsf{T}$-algebras of the form $F Y A$ for $A$ an object in $\mathbb{A}$ has the property that the restricted Yoneda embedding $\mathcal{B} \to [\mathbb{B}^\textrm{op}, \textbf{Set}]$ is fully faithful and preserves all colimits. But the universal property of $[\mathbb{B}^\textrm{op}, \textbf{Set}]$ implies that this is an equivalence of categories, so in fact every monad whose underlying endofunctor preserves colimits must be of this form.