Suppose $X,Y$ are locally compact Hausdorff spaces admitting a product $X\otimes Y$ in the category $\mathcal{H}$ of all such spaces. If one of $X,Y$ is empty, then $X\otimes Y$ exists, so I'll assume that neither is.
Then the points of $X\otimes Y$ are exactly the maps $\ast\rightarrow X\otimes Y$. Since $X\otimes Y$ is Hausdorff, any such map is proper. Because of this, both of the categorical projections $$X\xleftarrow{\pi_X} X\otimes Y\xrightarrow{\pi_Y}Y$$ are surjective. They are continuous and proper by assumption.
Working now in $Top$, the category of all spaces, we have the Tychonoff product $X\times Y$ and an induced map $$\theta:X\otimes Y\rightarrow X\times Y$$ factoring the projections. By the above observation, $\theta$ is surjective.
Lemma: Suppose that $f:A\rightarrow B$ and $g:B\rightarrow C$ are maps of spaces $A,B,C$. If $f$ is surjective and $g\circ f$ is proper, then $g$ is proper. $\quad\blacksquare$
Apply the lemma to the composite $$pr_X\circ \theta=\pi_X$$ to conclude that $pr_X$ is proper. This is only possible if $Y$ is compact. Similarly, $X$ must also be compact.
Of course, if $X,Y$ are compact, then the product $X\otimes Y$ exists in $\mathcal{H}$ and is given by the Tychonoff product $X\times Y$.
If $X,Y$ are locally compact Hausdorff and nonempty, then the product $X\otimes Y$ exists in $\mathcal{H}$ if and only if both $X,Y$ are compact.