Dividing users with certain files into 2 equal groups
The problem is NP-complete. The reduction is from the Set cover problem. Let one user has all the files $F$. Then our only hope is to achieve $\mathcal{F}(G_2)= \mathcal{F}(G_1)$. So the question is, from given $2k-1$ sets, are there $k$ that cover the ground set? This easily reduces to the decision version of the Set cover problem after adding a few dummy users/files.

My old answer, which gives a polytime algorithm if we do not allow $\mathcal{F}(G_2)= \mathcal{F}(G_1)$:
Yes, you can check this in P. Your condition is equivalent to that there is an $f_i$ and $f_j$ such that $f_i\in \mathcal{F}(G_1)\setminus \mathcal{F}(G_2)$ and $f_j\in \mathcal{F}(G_2)\setminus \mathcal{F}(G_1)$. So all you have to do is for each pair, $f_i, f_j$, you check whether you can divide the users appropriately. This can be done if no user has both files and at most $k$ users have any of the files. The total running time is $O(n^2k)$ (if the info is stored appropriately) which can be easily improved.