On Wikipedia, we find the following remark without reference: (here $\pi:E \to X$ is the espace étalé of a (pre)sheaf on $X$)
It is possible to turn $E$ into a scheme and $\pi$ into a morphism of schemes in such a way that $\pi$ retains the same universal property, but $\pi$ is not in general an étale morphism because it is not quasi-finite. It is, however, formally étale.
So I'm wondering: what is the scheme structure on $E$? Do you we need some assumptions on the sheaf, such as quasicoherence?