The "satchel, signals" was used for all sorts of jobs, it was quite well padded and used to store radio headsets in armoured vehicles, but I have never seen proof that they carried messages in them?
In an early-war context, it's rare to see a Signals DR with anything more than his respirator case slung. Later on of course, the machines were fitted with panniers.
In action, despatches were often single-sheet messages and there are a couple of photos which seem to show them simply being put inside the jacket.