Parallel merge sort using hypercube connection template
They write


  The $i$-th parallel merge takes two sequences, each distributed over $2^{i-1}$ tasks, and generates a sorted sequence distributed over $2^i$ tasks.


For $i=1$, that reads


  The first parallel merge takes two sequences, each stored locally on one task, and generates a sorted sequence distributed over two tasks.


That is, every node performs one compare-merge. Similarly, for higher $i$, each node performs $i$ such operations.

They use it to derive "per-processor communication cost" $T_{comm}$ which clarifies that they do indeed count processor-wise in the prequel. That makes sense, too: communication happens in parallel, so in order to derive bounds on overall runtime you can bound every (abstract) step from above with the longest processing time found among all processors in that step.