My answer Wiener measure and Bochner Minlos can help defining a free massless real scalar field as a random element of $S_0'(\mathbb{R}^2)$. Namely, take the Schwartz space of rapidly decaying test functions $S(\mathbb{R}^2)$ and consider the subspace $S_0(\mathbb{R}^2)=\{f| \widehat{f}(0)=0\}$ of "charge neutral" test functions. The bilinear form $$ B(f,g)=\int_{\mathbb{R}^2} \frac{d^2\xi}{(2\pi)^2}\ \frac{\overline{\widehat{f}(\xi)}\widehat{g}(\xi)}{|\xi|^2} $$ is continuous and positive on $S_0(\mathbb{R}^2)$ and therefore by the Bochner-Minlos Theorem there exists a unique centered Gaussian probability measure $\mu$ on the topological dual $S_0'(\mathbb{R}^2)$ for which $$ \mathbb{E}(\phi(f)\phi(g))=B(f,g) $$ where $\phi$ is the corresponding random element in $S_0'(\mathbb{R}^2)$.
I did not do the computation (which needs a lot of care) but I suspect that $B(f,g)$ should be a multiple of $$ \int_{\mathbb{R}^2\times\mathbb{R}^2} d^2 x\ d^2y\ f(x)(-\log|x-y|)g(y)\ . $$ The issue here is that you have both a UV and an IR problem to deal with. The slow decay of the propagator for large $\xi$ makes it so that $X$ or $\phi$ is a random distribution rather than a random function and, in particular, punctual evaluations $X(x)$ do not make sense. Also, the divergence at $\xi=0$, or zero-mode, makes it so that there is an ambiguity of shifting the field by a constant so the field $X$ does not make sense but its "increments" do. Working with $\partial X$ is another way to circumvent this issue.