# How to find the joint distribution of sums of Poisson random variables

I am trying to determine the joint distribution of two sums of Poisson random variables.

Let's say $X \sim \text{Pois}(\lambda_{1})$, $Y \sim \text{Pois}(\lambda_{2})$, and $Z \sim \text{Pois}(\lambda_{3})$. Moreover assume that $X,Y$ and $Z$ are all mutually independent. I want to find $P(X=k|X+Y=n_{1}, X+Z=n_{2})$.

From this step, I know $P(X=k|X+Y=n_{1}, X+Z=n_{2}) = \frac{P(Y=n_{1}-k, Z= n_{2}-k) P(X)}{P(X+Y, X+Z)}$.

I know the numerator is a bivariate Poisson distribution and marginal Poisson. How would I go about finding the distribution for the denominator?

Sum over all possible values of $X$. You can sum because of countable additivity of $P(\cdot)$. The goal is to exploit the independence among your three random variables (note: I suggested this addition to your problem description in an edit just now...without this my answer is incorrect.).
Also, you can further simplify the numerator by factoring the first factor into the two pmfs of $Y$ and $Z$.