Do you have to run nodetool repair on every node? Do you have to run nodetool repair on every node in a cluster, or do you only need to run it on one node, and from there Cassandra will take care of the rest?
In looking through the documentation, I am having a hard time finding a reference for this, but the short answer is "yes" you need to run nodetool repair on each node in your cluster.  The closest I can find is the documentation on repairing nodes which suggests that you should not run repair on more than one node in your cluster at a time.

You can also run repair with the -pr flag, which limits the repair operation to the first token range that current node is responsible for.  This cuts down on the duplication of work when run on the remaining nodes.