Put $m=n+1$. Let $X,Y$ be the sets of prefix sums of the two sequences, including zero. Each has $m$ elements.

A collision $x+y=x'+y'$ between distinct pairs is equivalent to a common positive difference in $X,Y$: after exchanging the pairs, $x-x'=y'-y>0$. These differences are exactly nonempty segment sums.

Thus all $m^2$ sums are distinct. Their maximum is

$$\max X+\max Y=n(n+2)=m^2-1,$$

so they represent every integer in $\{0,\ldots,m^2-1\}$ exactly once.

Conversely, such sets give positive sequences with disjoint segment sums by taking consecutive differences of their sorted elements. We must also check that neither sequence is constant.