Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. A Poisson bivector on $X$ is defined by \begin{align} \pi_X = \sum_{i,j} \{x_i, x_j\} \partial_{x_i} \wedge \partial_{x_j}, \end{align} where $\{,\}$ is a Poisson bracket on $C^{\infty}(X)$. Similarly, we have a Poisson bivector on $Y$: \begin{align} \pi_Y = \sum_{i,j} \{y_i, y_j\} \partial_{y_i} \wedge \partial_{y_j}, \end{align} where $\{,\}$ is a Poisson bracket on $C^{\infty}(Y)$. My question is: how to define $\pi_{X \times Y}$? I think that a local coordinate on $X \times Y$ is $(U \times V, x_1 \otimes y_1, \ldots, x_n \otimes y_m)$. Maybe \begin{align} \pi_{X \times Y} = \sum_{i,j,s,t} \{ x_i \otimes y_j, x_s \otimes y_t \} \partial_{x_i \otimes y_j} \wedge \partial_{x_s \otimes y_t}. \end{align}
But I don't know how to compute $\partial_{x_i \otimes y_j}$. Are there some relation between $\pi_X$, $\pi_Y$ and $\pi_{X \times Y}$? Thank you very much.
By definition, the Poisson product manifold of $(X,\pi_X)$ and $(Y,\pi_Y)$ is the manifold $$(X\times Y,\pi_X+\pi_Y).$$ For a reference, see for example Lectures on the geometry of Poisson manifolds by Izu Veisman.