Disclaimer: these are my musings about what's going on, without actually having seen anything that properly explains things.
First the stuff I do know. Let $V^*$ denote the space of all linear functionals on a vector space $V$.
An important part of multilinear algebra is the tensor product. You can look this up, but the key idea is that $V \otimes W$ is the target space for the most general way for multiplying vectors from $V$ with vectors from $W$ to get a result that is still a vector space, and such that the corresponding tensor product of vectors $\otimes : V \times W \to V \otimes W$ is a bilinear function.
If $V$ and $W$ are finite dimensional, and $v_i$ and $w_j$ are bases, then a basis for $V \otimes W$ would be given by the set $v_i \otimes w_j$.
The odd thing about multilinear algebra is that things can be combined in a lot of ways. For example, a linear functional $T : V \to \mathbf{R}$ can be used to construct a map $V \otimes W \to W$, defined on a generating set by the formula
$$ T(v \otimes w) = T(v) w $$
Now, the stuff I don't know.
I assume $\mathscr{S}(\mathbf{R}^n)$ denotes the space of test functions. Since the ordinary product of a test function in $x$ and a test function in $y$ is a bilinear map, there is a corresponding linear transformation
$$ \mathscr{S}(\mathbf{R}) \otimes \mathscr{S}(\mathbf{R}) \to \mathscr{S}(\mathbf{R}^2) $$
which replaces the tensor product with the ordinary product. I believe this map is continuous, injective, and has dense image.
For two linear functionals $S$ and $T$ on $\mathscr{S}(\mathbf{R})$, their tensor product acts on the space of tensor products of test functions, given by the formula on a generating set:
$$ (S \otimes T)(f \otimes g) = S(f) T(g) $$
We can thus extend $S \otimes T$ by continuity to be a partial linear functional on $\mathscr{S}(\mathbf{R}^2)$.
And this is about where my musings peter out. Maybe $S \otimes T$ is always a totally defined functional? In any case, a key point is that I'm not trying to convolve two arbitrary distributions on $\mathbf{R}^2$: instead, I'm trying to find a decomposition where I can split the problem into separate variables so that the two distributions are univariate.
This would all be nicer with Hilbert spaces; above when I say "tensor product", I mean the tensor product of the vector space structure. I think the tensor product of the Hilbert space structure works out to be nicer, so that we actually have an isomorphism $\mathscr{L}(\mathbf{R}) \otimes \mathscr{L}(\mathbf{R}) \cong \mathscr{L}(\mathbf{R}^2)$ as well as an isomorphism $H^* \cong H$, and all the facts I know about multilinear algebra still apply too.