I've noticed that often authors will comment that "descent theory" shows that some sheaf in the étale topology is actually a scheme. I was wondering what result in descent theory actually implies this(a statement and reference for the result is what is requested in case an actual explanation would take too long). I understand how descent theory is what allows one to define and construct stacks, but I've never seen a result which says something to this effect (for example, I haven't seen it in Vistoli's stack notes in FGA explained) and it seems to be used often in verifying that something is a DM stack.

For example, Edidin states in the proof of Prop. 2.2 of his notes on $M_g$ that `"$\underline{\mathrm{Iso}}_B(e,e')$`

is the étale sheaf which is the quotient of $(X\times_{X\times X} E\times_B E')$ by the free group action of $G$. Moreover, descent theory shows that this sheaf is in fact a scheme."

In the proof of Theorem 3.2 there, he similarly writes, "Descent theory says that in this case a quotient $C=C_E/PGL(N+1)$ also exists as a scheme."

One finds similar statements in the other standard papers about DM stacks. I was wondering what results in "descent theory" they are referring to.