Define a homomorphism $\varphi : A \to B$ of commutative discrete rings or commutative ring spectra to be a (effective) descent morphism if the comparison functor from $\mathsf{Mod}_A$ to the category of descent data for $\varphi$ is fully faithful (is an equivalence). In the discrete case it's known that $\varphi$ is a descent morphism iff it's an effective descent morphism iff it's pure. As far as I know there isn't an analog of the last condition for spectra which is known to imply descent. But are the first two still equivalent? Is a descent morphism of $\mathbb{E}_{\infty}$ ring spectra also an effective descent morphism?
Edit: The number of edits on this post was getting too long and so I've deleted them & summarize the main points here (see post history for more detail). The comparison functor $\kappa : \mathsf{Mod}_A \to \operatorname{Desc}(\varphi)$ does not preserve compact objects in the derived setting (unlike in the discrete setting); this and related facts like the limit expressing descent being infinite make me think purity in the sense of triangulated categories would not suffice to capture descent.
Note $\varphi$ is a descent morphism iff every $A$-module is $B$-nilpotent complete. Every $A$-module is $B$-nilpotent complete iff every $A$-algebra is $B$-nilpotent complete, because of the trivial square zero extension construction. So $\varphi$ is a descent morphism for modules iff it is a descent morphism for algebras (iff it is a universal effective monomorphism in the category of commutative ring spectra). If $\varphi$ is an effective descent morphism for modules then it is such for algebras because the functor $\operatorname{CAlg}(-)$ from symmetric monoidal $(\infty, 1)$-categories to $(\infty, 1)$-categories preserves limits (it is corepresentable in the $(\infty, 2)$-sense). The converse isn't clear to me but would follow if the formation of Beck modules preserved limits.
If that converse is true then we are then ultimately asking for a weak kind of Barr-exactness of $\mathsf{Aff} = \operatorname{CAlg}(\mathsf{Sp})^{\mathrm{op}}$. We want to know that the self-indexing/codomain fibration is a stack with respect to the topology generated by singleton covers consisting of an universal effective epimorphisms (we know it's a prestack, i.e. separated). This is strange because $\mathsf{Aff}$ is far from being regular, the only clear exactness property is has is extensivity, so maybe it's just a coincidence that things work out in the discrete case. None of the proofs I have seen of this fact in the discrete case seem to generalize well.
