I believe that Dennis makes a key point: a GEC follows different rules than an EGC.
These rods are already connected together by EGCs. However only the rod directly connected to a building via a GEC counts as a grounding electrode for that building.
I don't see any prohibition from using one electrode to serve multiple structures, as long as suitable GECs are providing the connection.
It is not clear to me, for the particular situation described by the original poster, that running 20' for conductor, _buried to the proper depth_, would in fact be easier than adding an additional ground rod.
On the other side of the coin, it is known that increased spacing of ground electrodes lowers resistance; ground electrodes that are too close together share the same 'sphere of influence' and thus are less effective. (Consider the limit of two electrodes side by side; they act as a single somewhat fatter electrode.) Using the 'shared electrode' approach, the result is electrodes that will be better spaced than adding a new electrode at each location.
-Jon