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We present SLASH (Sketched LocAlity Sensitive Hashing), an MPI (Message Passing Interface) based distributed system for approximate similarity search over terabyte scale datasets. SLASH provides a multi-node implementation of the popular LSH (locality sensitive hashing) algorithm, which is generally implemented on a single machine. We show how we can append the LSH algorithm with heavy hitters sketches to provably solve the (high) similarity search problem without a single distance computation. Overall, we mathematically show that, under realistic data assumptions, we can identify the near-neighbor of a given query $q$ in sub-linear ($ \ll O(n)$) number of simple sketch aggregation operations only. To make such a system practical, we offer a novel design and sketching solution to reduce the inter-machine communication overheads exponentially. In a direct comparison on comparable hardware, SLASH is more than 10000x faster than the popular LSH package in PySpark. PySpark is a widely-adopted distributed implementation of the LSH algorithm for large datasets and is deployed in commercial platforms. In the end, we show how our system scale to Tera-scale Criteo dataset with more than 4 billion samples. SLASH can index this 2.3 terabyte data over 20 nodes in under an hour, with query times in a fraction of milliseconds. To the best of our knowledge, there is no open-source system that can index and perform a similarity search on Criteo with a commodity cluster.