The answer is given in the following - very interesting - paper:
 
> Bachuki Mesablishvili, *Descent Theory for Schemes*, Applied Categorical Structures 12: 485–512, 2004.
 
The author generalizes Grothendieck's descent theory from faithfully flat morphisms to *pure* morphisms. These are morphisms of schemes $f$ such that every base change $f'$ of it is schematically dense in the sense that $f'$ does not factor through a proper subscheme. In the affine case, $\mathrm{Spec}(A) \to \mathrm{Spec}(R)$ is pure iff $R \to A$ is "stable injective" in the sense that for every $R$-algebra $B$ the map $B \to A \otimes_R B$ is injective.
 
One of the main results (Theorem 5.15) states that for a quasi-compact morphism $f : X \to Y$ the following are equivalent:

1) $f$ is pure

2) $f^\* : \mathrm{Qcoh}(Y) \to \mathrm{Qcoh}(X)$ faithful

3) $f$ is a stable effective descent morphism for quasi-coherent sheaves
 
In the case that $X,Y$ are affine there are even more characterizations (Theorem 4.18), for example we can also add 4) $f^\*$ is conservative, and 5) $f^\*$ is comonadic.