There has been considerable recent interest in distribution-tests whose run-time and sample requirements are sublinear in the domain-size $k$. We study two of the most important tests under the conditional-sampling model where each query specifies a subset $S$ of the domain, and the response is a sample drawn from $S$ according to the underlying distribution. For identity testing, which asks whether the underlying distribution equals a specific given distribution or $\delta$-differs from it, we reduce the known time and sample complexities from $O(\delta^{-4})$ to $O(\delta^{-2})$, thereby matching the information theoretic lower bound. For closeness testing, which asks whether two distributions underlying observed data sets are equal or different, we reduce existing complexity from $O(\delta^{-4} \log^5 k)$ to an even sub-logarithmic $O(\delta^{-5} \log \log k)$, and providing a better bound to an open problem in Bertinoro Workshop on Sublinear Algorithms.