Let us work in the "nice" situation where $X,Y,Z$ are smooth complex algebraic varieties, not necessarily compact. Assume that the fiber product $W:= X \times_Z Y$ is also smooth. What can we say about the Picard group of $W$?
More precisely, assume that $Pic(X)=Pic(Z)=0$.
1) Can we deduce that $Pic(W)=Pic(Y)$?
2) If 1) is not true in general, can we draw the conclusion if $Z$ is a point and thus $W=X \times Y$?
3) If 1) is not true in general, can we draw the conclusion if $Y \to Z$ is a finite étale cover?