That the probability is at least $O(1/n^2)$ is immediate, since the probability that both permutations are $n$-cycles is $1/n^2.$ For the upper bound, there is a [convergence speed estimate by Zacharovas][1], see in particular theorems 3 and 4, which should give you what you want.


  [1]: http://arxiv.org/abs/0901.1733