The elements of coalgebras can often be thought of as having a distribution-like character, as can be seen from various examples. For example, the cofree coalgebra cogenerated by a single element (let us say over a field $k$ of characteristic zero) can be realized as a localization of $k[x]$ at the prime ideal $(x)$, which we can think of as sitting inside a space of formal power series
$$k[x]_{(x)} \hookrightarrow k[[x]]$$
by expanding inverses of elements prime to $x$ in geometric series. A way to think about the formal series $\sum_n a_n x^n$ that so arise is that $x^n$ is a symbol for a derivative of a Dirac functional in the sense of distributions, i.e., we think of a formal series as a formal sum
$$\sum_n a_n \frac{\delta_{0}^{(n)}}{n!}$$
where $\delta_0 = eval_0: k[x] \to k$ is the Dirac functional at $0$. The comultiplication can be read off from the adjoint rule
$$\langle \Delta(m), f \otimes g\rangle = \langle m, f \cdot g\rangle$$
where $f, g \in k[x]$ are polynomial "functions". I wrote up a calculational exposition of this point of view on the cofree coalgebra at the $n$-Category Café here.
A general heuristic here is that it's hard to multiply distributions, but one can often "comultiply" them by this adjoint rule.
Another example is given by homology theory. Here the adjoint pairing (say if we are thinking in terms of De Rham theory, where cocycles are given by smooth functions) is given by integration of an $n$-form over an $n$-chain:
$$\langle c, \omega \rangle = \int_c \omega$$
so here we are thinking of $n$-chains as acting on $n$-forms much as currents. In this way, the homology coalgebra also has a distributional character (with the same sort of interpretation of comultiplication).