Let $T \neq 0$ be a finite collection of 'sites'. A random field $X$ on $T$ with values in $L$ is a random vector $(X_i)_{i \in T}$ having $L$-vlaue components. If $L$ is finite, the distribution of $X$ is specified by the probability mass function \begin{align} \pi_X(x) = \mathbb{P}(X_i = x_i : i \in T), \qquad x \in L^T, \end{align} and by a joint density function $\pi_X$ in the continuous case $L \subseteq \mathbb{R}$.
The local characteristics of a random field $X$ are \begin{align} \pi_i(x_i | x_{T \setminus i})=\pi_X(X_i =x_i | X_{T \setminus i} = x_{T \setminus i}) \end{align} whenever well-defined.
Now, I want to show that the local characteristics \begin{align} \pi_1(x|y) = \pi_2(y|x) = \frac{1}{\sqrt{2\pi}}\exp\big[-\frac{1}{2}(x-y)^2\big] \end{align} are well-defined but do not define a proper joint distribution. Do I have to show that \begin{align} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}\exp\big[-\frac{1}{2}(x-y)^2\big]\ dx\ dy \neq 1 ? \end{align} How to show the well-definedness?
No, the conditional probability density function should not be inegrate to 1. Let local characteristics define a proper joint distribution $\pi_{1,2}(x,y)$. Then
$$\pi_1(x\mid y) = \frac{\pi_{1,2}(x,y)}{\pi_2(y)}, \quad \pi_2(y\mid x) = \frac{\pi_{1,2}(x,y)}{\pi_1(x)}. $$ L.h.s.'s of this equalities are the same. Then r.h.s.'s are equal: $$\frac{\pi_{1,2}(x,y)}{\pi_2(y)}= \frac{\pi_{1,2}(x,y)}{\pi_1(x)} \iff \pi_1(x)=\pi_2(y) \text{ for any } x,y. $$ This cannot be true.
To show the well-definedness of $\pi_1(x\mid y)$, check whether it is a p.d.f. You could know what distribution has such p.d.f.