Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 1$ and
\[f(x^2 - y^2) = (x - y) (f(x) + f(y))\]for all real numbers $x$ and $y.$

Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$  Find $n \times s.$
Setting $x = y,$ we get $f(0) = 0.$

Setting $x = -1$ and $y = 0,$ we get
\[f(1) = -f(-1),\]so $f(-1) = -1.$

Setting $y = 1$ and $y = -1,$ we get
\begin{align*}
f(x^2 - 1) &= (x - 1) (f(x) + 1), \\
f(x^2 - 1) &= (x + 1) (f(x) - 1),
\end{align*}respectively.  Hence, $(x - 1) (f(x) + 1) = (x + 1) (f(x) - 1),$ which simplifies to $f(x) = x.$  We can check that this function works.  Therefore, $n = 1$ and $s = 2,$ so $n \times s = \boxed{2}.$