The post contains some elements of a correct approach.
We cannot solve the problem without making some assumptions about the relationship between $X_1$ and $X_2$. We will assume that $X_1$ and $X_2$ are independent.
Let $S=X_1+X_2$. We want to find $\Pr(S=n)$. This is $0$ if $n\le 1$. So let $n\ge 2$.
We can have $S=n$ in several ways. For we could have $X_1=1$ and $X_2=n-1$. Or we could have $X_1=2$ and $X_2=n-2$. And so on up to $X_1=n-1$ and $X_2=1$.
For any $k$ from $1$ to $n-1$, we have
Adding up from $k=1$ to $k=n-1$ we find that
Remark: We sketch another approach. We can think of $X_1$ as the number of trials until the first success, and of $X_2$ as the number of trials between the first success and the second success. So $S$ is the number of trials until the second success. The probability this is $n$ is the probability of exactly one success in the first $n-1$ trials times the probability of success on the $n$-th trial. That gives
For related matters, please look under negative binomial distribution.