Vistoli's notes on fibred categories and descent - http://homepage.sns.it/vistoli/descent.pdf - introduce (section 4.2.1) descent on modules over a commutative ring. The idea is as follows:
Define a pseudofunctor on the category of commutative rings that sends a ring $A$ to the category $\mathbf{Mod}_A$ of modules over $A$ and sends a ring homomorphism $f:A\to B$ to the extension of scalars functor $M\mapsto B_f\otimes_A M$ for $M$ an $A$-module (where we make $B$ into an $A$-module $B_f$ via $f$).
Form the associated fibred category via the Grothendieck construction.
Equip the category $\mathbf{Rings}$ of commutative rings with the faithfully flat topology - that is, the topology given by the pretopology on $\mathbf{Rings}$ whose coverings consist of a single faithfully flat homomorphism - so they are of the form $\{f:A\to B\}$, where $f$ is faithfully flat.
The descent data are then isomorphisms of $B\otimes B$-modules between the two modules $B\otimes M$ and $M\otimes B$, and isomorphisms of $B\otimes B\otimes B$-modules between the three modules $B\otimes B\otimes M,B\otimes M\otimes B, M\otimes B\otimes B$ satisfying the usual compatibility and cocycle conditions. Given these, we can show that we have defined a stack over $\mathbf{Rings}$.
But what if, at the start, we define our fibred category slightly differently, constructing it from the pseudofunctor that sends $A$ to $\mathbf{Mod}_A$ as before, but which now sends $f:A\to B$ to the functor $N\mapsto N_f$, where $N$ is a $B$-module, and $N_f$ is the $A$-module obtained from $N$ by restriction of scalars along $f$. Since restriction and extension of scalars are adjoint, I'd expect our new category to be basically the same.
Another interesting point is that what we have defined is actually a functor, not a pseudofunctor, since we have actual equalities $N=N_{\textrm{id}}$ and $\left(P_f\right)_g=P_{fg}$ rather than the canonical isomorphisms $M\cong M\otimes_A A$ etc. The distinction isn't important in practice, but it makes things a lot clearer.
Another difference is that this functor is now contravariant. But the (pseudo)functors seen in descent theory often are contravariant, so this isn't a problem.
What is a problem is that tensor products don't appear at all in our new definition, and I can't see how to define descent data. When we used extension of scalars we had interesting different ways to make an $A$-module $M$ into a $B$-module along a map $f:A\to B$ - take $B_f\otimes_A M$ or $M\otimes_A B_f$. Similarly for the triple products above. But there's nothing like that when working with restriction of scalars.
Can it still be made to work?