In this work, we study the $k$-median and $k$-means clustering problems when the data is distributed across many servers and can contain outliers. While there has been a lot of work on these problems for worst-case instances, we focus on gaining a finer understanding through the lens of beyond worst-case analysis. Our main motivation is the following: for many applications such as clustering proteins by function or clustering communities in a social network, there is some unknown target clustering, and the hope is that running a $k$-median or $k$-means algorithm will produce clusterings which are close to matching the target clustering. Worst-case results can guarantee constant factor approximations to the optimal $k$-median or $k$-means objective value, but not closeness to the target clustering. Our first result is a distributed algorithm which returns a near-optimal clustering assuming a natural notion of stability, namely, approximation stability [Balcan et. al 2013], even when a constant fraction of the data are outliers. The communication complexity is $\tilde O(sk+z)$ where $s$ is the number of machines, $k$ is the number of clusters, and $z$ is the number of outliers. Next, we show this amount of communication cannot be improved even in the setting when the input satisfies various non-worst-case assumptions. We give a matching $Ω(sk+z)$ lower bound on the communication required both for approximating the optimal $k$-means or $k$-median cost up to any constant, and for returning a clustering that is close to the target clustering in Hamming distance. These lower bounds hold even when the data satisfies approximation stability or other common notions of stability, and the cluster sizes are balanced. Therefore, $Ω(sk+z)$ is a communication bottleneck, even for real-world instances.