A listener in Huntsville, Alabama, says that in her native Scotland, the phrase send out for messages means to send someone to go shopping. The phrase stems from a time when the person going out to do the shopping or run other errands would also pick up the postal mail, sometimes at the local store. This is part of a complete episode.
Not sure I agree with the derivation suggested; surely as the goods are your groceries, it is likely that the connection is to the βmessβ of βmess hallβ, βmess of pottageβ, etc., and is to do with the fact itβs foodstuffs? Iβm sure that I was told the root was Old French βmesβ, a βportion of foodβ. This sense does derive ultimately from the same Latin which gives us βmessageβ and βmessengerβ in their communications meanings, but I think that the distinction happened earlier than you suggested in the programme.
Oh, and you *can* βrunβ or βgo a messageβ (singular) to indicate a brief errand/ shopping trip, at least in NE Scotland where I come from, and probably elsewhere in Scotland too.