Let $X=\mathbb{P}^1$. Then the naive definition of the Picard functor $Pic_X$ sending $S \to Pic(X \times S)$ fails to be a sheaf on the Zariski topology because any line bundle on $Pic(X \times S)$ will trivialize along $Pic(\mathbb{A}^1 \times S)$.
So, instead we redefine $Pic_X: coker(Pic(S) \to Pic(X \times S))$.
I don't see how this fixes things. Namely, let $\mathcal{L} \in Pic(X \times S)/f^*(Pic(S))$. I am not entirely sure but it seems that there should still exist non-trivial $\mathcal{L}$.
Anyway, we can still consider the Zariski covering $\mathbb{A}^1_S \times \mathbb{A}_S^1$ then the pullback of $Pic_X(S)$ to each open subset in this cover should be a line bundle ino $Pic(\mathbb{A}^1_S)/f^*Pic(S))$.
But all of these are still trivial, and so, in particular, $Pic_S$ would still fail to be a sheaf on the Zariski topology.
Here $f=pr: X \times S \to S$ and also $\mathbb{A}^1_S \to S$.