Assuming the two subsets are selected independently, the probability in question is 
$$p_n=s_n/2^{2n},$$
where $a_n$ the sum of the squares of the coefficients in the polynomial
$$\prod_{k=1}^n (1+x^k).$$
The sequence $(a_n)$ is the sequence [A047653][1], and its asymptotics, given on that page, is 
$$a_n\sim\frac{\sqrt{3/\pi}\,4^n}{n^{3/2}}$$
(as $n\to\infty$). So, 
$$p_n\sim\frac{\sqrt{3/\pi}}{n^{3/2}}.$$


  [1]: https://oeis.org/A047653