Suppose I have two random partitions of $N.$ ("random" really means "the cycle type of a random permutation", but if there is an answer with any definition, I am interested). The question is: what is the probability is that no subset sum of the first partition is equal to any subset sum of the second partition? I seem to have trouble finding references...

Here are two trivial observations while we wait for the real experts to completely solve this problem (paging Prof. Stanley...) First, note that there is a reformulation of this question that might appeal more directly to partition theorists. Let $P(n)$ be the set of all partitions of $n$. Given partitions $\pi \in P(n)$ and $\rho \in P(m)$, we write $\rho \subset \pi$ to indicate containment as multisets, meaning that each part of $\rho$ is also a part of $\pi$. Note that if $\rho \subset \pi$ then we must have $m \leq n$ for obvious reasons. For each $n$, let $\Gamma(n)$ be the set of partitionpairs $(\pi_1,\pi_2) \in P(n) \times P(n)$ for which there exists a triple $(m,\rho_1,\rho_2)$ with $m < n$, $\rho_j \in P(m)$ and $\rho_j \subset \pi_j$. This question is asking:
Secondly, one can compute $\Gamma(n)$ by hand for small $n$. It helps a lot, at least for tiny $n$, to note that $\Gamma(n) \geq P(n1)^2$: the latter quantity counts those partitions of $n$ which contain $1$, so no two such partitions can hope to be subsetindependent. Here are some values resulting from my (possibly erroneous) calculations: $$\Gamma(2) = 2,\\ \Gamma(3) = 5,\\ \Gamma(4) = 15, \\ \Gamma(5) = 35.$$ I am not happy with the unique oeis entry for this sequence, but you might find the $20$ candidates for $P(n)^2  \Gamma(n)$ much more promising, see here. 

