Efficient algorithm for testing planarity of the union of two planar graphs
You might be interested in an online algorithm, which is one that processes the input in a serial manner. Here is a paper with fast algorithms that tests planarity after adding an edge or a vertex, and does so much faster than $O(m+n)$ time (it does so in $O(\log{n})$):

http://epubs.siam.org/doi/abs/10.1137/S0097539794280736?journalCode=smjcat

You will be interested in the edge-addition version of this. And you could ask your question with any two edge sets, but since you seem to have info that both $G_1$ and $G_2$ are planar, you can start with the one which is larger, knowing it is planar, then add edges from the other graph one at a time.

In a 'worst case' where $|E_1| = |E_2| = m$, you would have to decide when this $\frac{m}{2}O(\log{n})$ is better or worse than just applying a $O(m+n)$ planarity test.