Since radios were bulky, expensive, unreliable and could be tapped into, much lower level communication (platoon to company, or even higher up in lower tech armies such as the Soviet Union's) within units was done through messengers bringing written notes from one officer to another. This was especially common on more mobile front lines where field telephone lines hadn't been set up or had been cut. It could be an incredibly dangerous job, as they didn't have the luxury of hunkering down in one place and were often in the open.

Sources: Stephen Bull, World War 2 Infantry Tactics (2) Company and Battalion