Assuming there is a set $S$ that you are given subsets of, $s_1, s_2, ..., s_n$, estimate $|S|$ (and a confidence interval if possible) making as few assumptions as possible.
I'm not going to quibble over assuming the subsets are selected in an independent or uncorrelated way.
If you can assume every element in $S$ has an equal chance of being selected and $n=2$:
$|S| \simeq \frac{|s_1| |s_2|}{|s_1 \cap s_2|}$
(Think of $s_1$ and $s_2$ as the lower and left portion of a square.)
But what about $n>2$ or each element has a different (but non zero) chance of being selected?