I will try to sketch something I know or collected from elsewhere. We know "regular$\Rightarrow$complete intersection$\Rightarrow$Gorestein$\Rightarrow$Cohen Macaulay". In thm 2 link text^{1} it is shown when A and B are E algebras, A is flat over E, B is finitely generated over E, then:

If A is a complete intersection, B and E are regular, then $A\otimes_E B$ is a complete intersection.

If A, B, E are Gorestein, then $A\otimes_E B$ is Gorestein.

It is proved in EGA when A, B, (also E, but this may be superfluous) are Cohen Macaulay then so is $A\otimes_E B$.

When E is a field k, B is finitely generated, the above hypothesis is satisfied, therefore our question is "almost correct". Now in my old notation as in the original question, suppose X and Y are geometrically integral, therefore the function field of Y $K_Y=Frac(B)$ is linearly disjoint from the perfect closure of $k$, i.e. $k^{1\over {p^{\infty}}}$. ~~Now by analogy with the "genus drop" phenomenon, see e.g.link text~~^{2}, I suspect (caveat!) $A\otimes_k Frac(B)$ is regular, in particular normal. Similarly $Frac(A)\otimes_k B$ is also regular, now $A\otimes_k B=A\otimes_k Frac(B)\cap Frac(A)\otimes_k B$ is normal.(Edit: fiber product of normal varieties is normal just by universal property, this part is superfluous) However we will show it is not regular.

Take the example as suggest by Qing Liu's comment, i.e. $A=B=k[S,T]/(T^2-S^p-t)$, $X=Spec A$, Y=$Spec B$, $x_0\in X$ is the maximal ideal generated by $(T)$, then there is a (in this case unique) maximal ideal in $A\otimes_k B$ containing $(T)\otimes_k B$ and $A\otimes_k (T)$, call it the point $(x_0, x_0)$.

~~(Note in general given $x\in X$, $y\in Y$, I suspect we do not get the point $(x,y)\in X\times_k Y$ for free by the universal property of fiber product. The reason is that a "closed point" in a scheme corresponds to a morphism from a field to the scheme, rather than the other way around.) The residue field at $(x,y)$ will be the composite field (ambiguity of Galois conjugation) of the corresponding residue field $k_{X,x}$ and $k_{Y,y}$, which may not be linearly disjoint even when $Frac(A)$ and $Frac(B)$ are.~~

Localize $A\otimes_k B$ at $(x_0, x_0)$, call it C with maximal ideal $m$. We need a lemma from EGA IV 17.1.8:

Lemma: Suppose C is Noetherian local ring, with maximal ideal $m$, $t\in m$, the following is equivalent:

- C/tC is regular, and t is not a zero divisor of C;
- C is regular, $t\notin m^2$.

Take t to be $1\otimes_k T\in m\setminus m^2$, then $C/tC=$ some localization of $A\otimes \mathbb{F}_p(t^{1\over p})$ which is not regular at the maximal ideal $m/tm$. Q.E.D

However, I don't know if there is a more direct way to see why $(x_0, x_0)$ is not a regular point of $X\times_k Y$, I would appreciate any comment.

^{1} Kei-ichi Watanabe. Takeshi Ishikawa. Sadao Tachibana. Kayo Otsuka. "On tensor products of Gorenstein rings." J. Math. Kyoto Univ. 9 (3) 413 - 423, 1969. https://doi.org/10.1215/kjm/1250523903

^{2}John Tate: Genus Change in Inseparable Extensions of Function Fields. Proceedings of the American Mathematical Society, Vol. 3, No. 3 (Jun., 1952), pp. 400-40