Let $M=M_i \times M_j$. Then if $Exp^i_{\mathbf{p}_i}:T_{\mathbf{p}_i}M_i \to M_i$ and $Exp^j_{\mathbf{p}_j}:T_{\mathbf{p}_j}M_j \to M_j$ are the exponential maps on $M_i$ and $M_j$, respectively, then $\left(Exp^i_{\mathbf{p}_i}, Exp^j_{\mathbf{p}_j}\right)=:Exp_{(\mathbf{p}_i, \mathbf{p}_j)} : T_{(\mathbf{p}_i, \mathbf{p}_j)}M_i \times M_j \to M_i \times M_j$ is the exponential map on $M_i \times M_j$.
Is this statement true? The "feeling" is that it is and an argument might be that for any $\mathbf{v}\in T_{(\mathbf{p}_i,\mathbf{p}_j)}M_i\times M_j$ the action of the product exponential map is to trace a geodesic having the length of $\mathbf{v}$ and for which $\dot{\gamma}(0)=\mathbf{v}$. Of course, the component exp maps must be defined on the factors of the product tangent space (we can either assume geodesic completeness or amend the definition of the product exp map such that we use a product of neighborhoods in the tangent space domain for which both exp maps are defined).