Large, distributed data streams are now ubiquitous. High-accuracy sketches with low memory overhead have become the de facto method for analyzing this data. For instance, if we wish to group data by some label and report the largest counts using fixed memory, we need to turn to mergeable heavy hitter sketches that can provide highly accurate approximate counts. Similarly, if we wish to keep track of the number of distinct items in a single set spread across several streams using fixed memory, we can turn to mergeable count distinct sketches that can provide highly accurate set cardinalities. If we were to try to keep track of the cardinality of multiple sets and report only on the largest ones, maintaining individual count distinct sketches for each set can grow unwieldy, especially if the number of sets is not known in advance. We consider the natural combination of the heavy hitters problem with the count distinct problem, the heavy distinct hitters problem: given a stream of $(\ell, x)$ pairs, find all the labels $\ell$ that are paired with a large number of distinct items $x$ using only constant memory. No previous work on heavy distinct hitters has managed to be of practical use in the large, distributed data stream setting. We propose a new algorithm, the Sampling Space-Saving Set Sketch, which combines sketching and sampling techniques and has all the desired properties for size, speed, accuracy, mergeability, and invertibility. We compare our algorithm to several existing solutions to the heavy distinct hitters problem, and provide experimental results across several data sets showing the superiority of the new sketch.