Given a scheme $Z$ over $X$, one can consider the sheaf of "sections $X\to Z$". More precisely, that sheaf associates to every étale open $U\to X$ the set of commutative diagrams $$ \begin{matrix} & & Z \\\ & \nearrow & \downarrow \\\ U & \to & X \end{matrix} $$ The scheme $\tilde Z$ you are looking for is the espace étalé of the above sheaf, a highly non-separated scheme.

Given a scheme $Z$ over $X$, one can consider the sheaf of "sections $X\to Z$". More precisely, that sheaf associates to every étale open $U\to X$ the set of commutative diagrams $$ \begin{matrix} & & Z \\\ & \nearrow & \downarrow \\\ U & \to & X \end{matrix} $$ The scheme $\tilde Z$ you are looking for is the espace étalé of the above sheaf, a highly non-separated scheme.