Which path would the electrical current take?

From one of my favorite books, Code: The Hidden Language of Computer Hardware and Software by Charles Petzold:

Here, he explains how to setup switches and lightbulbs in two locations with the purpose of communicating by turning the lights on and off according to some previously established protocol/code.

My question: is he oversimplifying - because that isn't the point of this chapter - which route/path the current would take when both switches are closed? As in:

Would the wire, which I awkwardly marked blue, also become charged and/or carry current? Or would there be no current because the cathodes of the two adjacent batteries cancel out any difference/voltage in the "blue wire" that would otherwise cause a current?