I am reading Milne's *Étale cohomology*, III.4.

A twisted form of an object $Y$ (a scheme, a sheaf of modules, of algebras...) over a scheme $X$ is an object $Y'$ such that there exists a covering in some topology (let's say étale to fix the ideas), $(U_i \to X)$ such that $Y \times_X U_i \cong Y' \times_X U_i$ for all $i$. Then, the book says, any twisted form of $Y$ defines a cocycle in $\check{H}^1(X,\mathrm{Aut}(Y))$ as follows: let $f_i$ be the isomorpisms $Y \times_X U_i \to Y' \times_X U_i$, then the cocycle is given by $(\alpha_{ij}): Y \times_X U_i \times_X U_j \to Y \times_X U_i \times_X U_j$ where $\alpha_{ij} = f_i^{-1} \circ f_j$, which, in turn, constitutes the descent data that eventually allows to recover $Y'$. As far as I understand, the descent data does not have to be always effective in general, but in some cases it is. The book gives two examples when this is the case: Severi-Brauer varieties and vector bundles.

I would like to understand when the Cech cohomology classes are in bijective correspondnence with the (isomorphism classes of) twisted forms. So I have two questions:

what are the general criteria that the descent data as described above is effective?

what happens if the trivialising cover of a twisted form $Y'$ contains just one étale morphism $U_0 \to X$? It seems like the cocycle defined by $Y'$ is always trivial then ($\alpha_{00} = f_0^{-1} \circ f_0$), yet the form $Y'$ might be not isomorphic to $Y$.