According to the Schwartz Kernel Theorem and its variants, there are the canonical isomorphisms $$\mathcal{D}^{\prime} \left(X\right)\tilde\otimes \mathcal{D}^{\prime} \left(Y\right)\simeq\mathcal{D}^{\prime}\left(X\times Y\right),$$ $$\mathcal{E}^{\prime} \left(X\right)\tilde\otimes \mathcal{E}^{\prime} \left(Y\right)\simeq\mathcal{E}^{\prime} \left(X\times Y\right),$$ $$\mathcal{S}^{\prime} \left(\mathbb R^n\right)\tilde\otimes \mathcal{S}^{\prime} \left(\mathbb R^m\right)\simeq\mathcal{S}^{\prime} \left(\mathbb R^{n+m}\right),$$ where $E\tilde\otimes F$ is the completion of the space $E\otimes F$.
Roughly speaking, this follows from the fact that the corresponding spaces of test functions $\mathcal{D}$, $\mathcal{C}^{\infty}_c$, \mathcal{C}^{\infty}$, and $\mathcal{S}$ are nuclear Fréchet spaces, and one has the canonical isomorphisms $$E^{\prime}\tilde\otimes F^{\prime}\simeq \left(E\tilde\otimes F\right)^{\prime}\simeq B(E,F),$$ when L(E; F'),$$ provided that $E$ and $F$ are nuclear Fréchet spaces. (Here the duals carry the strong dual topology and the space $B(E,F)$ L(E;F ')$ of continuous linear mappings is endowed with the topology of uniform convergence on the products of bounded sets.convergence.)
As Johannes mentioned in his comment, a detailed presentation of the Schwartz Kernel Theorem and its versions for various spaces of distributions can be found in Topological Vector Spaces, Distributions and Kernels by Trèves. (More specifically, take a look at Chapt. 51, "Examples of Nuclear Spaces. The Kernels Theorem".)
