Both companies share many component suppliers, and an Airbus (or Airbus-related) division is making parts for the 787. Boeing is moving to outsourcing everything but final large-scale component assembly. The 787 will be the first model to be built this way, and Japan is capable of pre-fabbing a good deal of the 777 structure that is now currently assembled in the Everett plant, so that line down the road may start getting larger pre-fabbed structures.
I am far less familiar with Airbus, but I imagine their various plants also outsource components before assembly and then these pre-fabbed components are sent to TLS
for final assembly.