Let $X$ be the DeligneMumford compactification of $\mathcal{M}_{0,n}$. Suppose I have two (big) line bundles $L$ and $L'$ on $X$ and that I want to show that they are the same element of $Pic(X)$. Of course one can check if they have same intersections with the Fcurves, or just check that they coincide over an open subset of $X$ whose complement has codimension at least 2. Do you see any other reasonable way? Perhaps considering the maps induced by their global sections?
