$2$-fiber product is a scheme then map of stacks is representable

This is not true even if $\mathcal X$ is an Artin stack. For example, let $G$ be a smooth group scheme over the base $T$, and let $\mathbf BG$ be its classifying stack (the category of $G$-torsors fibered over $Sch/T$). Then $T\times_{\mathbf BG}T=G$ is a scheme. However, the atlas $T\to \mathbf BG$ is not representable by schemes in general. An example where $G$ is an elliptic curve over a normal local scheme $T$ of dimension 2 can be found in the article Ineffective descent of genus one curves by Wouter Zomervrucht (arXiv:1501.04304).

What is true is that if $X$ is an algebraic space, $f:X\to \mathcal X$ is an effective epimorphism of fppf sheaves of groupoids on $Sch/T$, and $X\times_{\mathcal X}X$ is an algebraic space, then $f$ is representable by algebraic spaces. Indeed if $U$ is any algebraic space and $U\to\mathcal X$ any map, then $U\times_{\mathcal X}X\to U$ is representable by algebraic spaces fppf-locally on $U$ (namely after base change to $U\times_{\mathcal X}X$, where it becomes a pullback of $X\times_{\mathcal X}X\to X$), hence it is representable by algebraic spaces because algebraic spaces satisfy fppf descent (which is a somewhat nontrivial fact, see https://stacks.math.columbia.edu/tag/04SJ).