We study the fundamental problems of (i) uniformity testing of a discrete distribution, and (ii) closeness testing between two discrete distributions with bounded $\ell_2$-norm. These problems have been extensively studied in distribution testing and sample-optimal estimators are known for them [17, 7, 19, 11]. In this work, we show that the original collision-based testers proposed for these problems [14,3] are sample-optimal, up to constant factors. Previous analyses showed sample complexity upper bounds for these testers that are optimal as a function of the domain size $n$, but suboptimal by polynomial factors in the error parameter $\epsilon$. Our main contribution is a new tight analysis establishing that these collision-based testers are information-theoretically optimal, up to constant factors, both in the dependence on $n$ and in the dependence on $\epsilon$.