Suppose we have a function $f(x;\theta)$ that depends on a parameter $\theta$. We also have another function $P(x, y;\theta)=f(x;\theta)\cdot f(y;\theta)$. For a fixed value of $\theta$, say $\theta_1$, the function $P(x, y;\theta_1)$ has a global maximum at $(x_1, y_1)$. However, for a different value of $\theta$, say $\theta_2$, the function $P(x, y;\theta_2)$ has a global maximum at $(x_2, y_2)$, which is not the same as $(x_1, y_1)$. Does there exist any function $f(x; \theta)$ that satisfies the condition $P(x_1, y_1;\theta_2)>P(x_1, y_1;\theta_1)$?
EDIT: Found a simple solution. $f(x; \theta) = m\theta \cdot e^{-m\theta(x - n(\theta))^2}$ or $X\text~N(n\theta, 1/m\theta)$ could be two such functions.
I take it that this inequality should hold for all $\theta_2\ne\theta_1$. (If not, you’d need to specify for which values of $\theta_2$ it’s meant to hold.)
If so, there can be no such function, since we’d have
$$ P(x_1,y_1;\theta_2)\gt P(x_1,y_1;\theta_1)\ge P(x_1,y_1;\theta_2)\gt P(x_2,y_2;\theta_2)\ge P(x_1,y_1;\theta_2)\;, $$
where the first strict inequality is your condition, the second strict inequality is the same condition with $1$ and $2$ swapped, and the weak inequalities are due to the fact that $(x_1,y_1)$ and $(x_2,y_2)$ are global maxima for the respective parameter values $\theta_1$ and $\theta_2$.