The Australian-English compound paper yabber denotes a letter (i.e., a written message).

The Australian-English word yabber:
– as a noun: denotes speech, language, talk;
– as a verb: means to talk.

This is the origin of yabber, according to Edward Ellis Morris (1843-1901 or 1902) in Austral English: A dictionary of Australasian words, phrases and usages, with those Aboriginal-Australian and Maori words which have become incorporated in the language and the commoner scientific words that have had their origin in Australasia (London: Macmillan and Co. Limited, 1898):

Yabber, n. Used for the talk of the aborigines. Some think it is the English word jabber, with the first letter pronounced as in German; but it is pronounced by the aborigines yabba, without a final r. Ya is an aboriginal stem, meaning to speak. In the Kabi dialect, yaman is to speak : in the Wiradhuri, yarra.

The earliest occurrences of yabber that I have found are, as a noun and as a verb, from Recollections of Bush Life in Australia, During a Residence of Eight Years in the Interior (London: John Murray, 1848), by the English cleric Henry William Haygarth (1821-1902):

Without any disparagement to the soft sex of other countries, the most loquacious of them all would bear little comparison with an Australian “gin,” ¹ when fairly moved to “yabber.”
[…]
I observed one of the tribe watching with great attention a stranger, who was holding forth with much volubility upon some current topic. When, in conclusion, he was asked what he thought of the speaker, he answered directly, “I believe no good that one.” It seemed that with them, as with us, the emptiest vessels make the greatest sound, and he persisted in his first opinion of one who, in his own phrase, had “too much yabber.”

¹ The offensive Australian-English noun gin designates an Aboriginal woman.

The noun occurs several times, as yabber-yabber, in The Eureka Stockade: The Consequence of Some Pirates Wanting on Quarter-deck a Rebellion ² (Melbourne: J. P. Atkinson and Co., 1855), by the Italian author and interpreter Raffaello Carboni (1817-1875)—for example in the following passage:

The learned Counsel spent a heap of dry yabber-yabber on the law of high-treason, to show its absurdity and how its interpretation had ever proved a vexation even to lawyers, then he tackled with some more tangible solids.

² This is what the National Museum of Australia says about the Eureka Stockade:

On 30 November 1854 miners from the Victorian town of Ballarat, disgruntled with the way the colonial government had been administering the goldfields, swore allegiance to the Southern Cross flag at Bakery Hill and built a stockade at the nearby Eureka diggings.
Early on the morning of Sunday 3 December, when the stockade was only lightly guarded, government troops attacked. At least 22 diggers and six soldiers were killed.

These are the earliest occurrences of paper yabber that I have found, in chronological order:

1-: From Round the Camp Fire: The Overlander’s Story, in Australian Sketches (Melbourne: Kemp & Boyce, 1888), by ‘Overlander’ (William Henry Thomes – 1824-1895):

There was not time to communicate beforehand with the stockman nearest to us (about thirty miles), and it was no good taking the blackboy with us, as he was not to be trusted, so I determined to send him off first with a paper yabber to the other stockman, telling him where we had gone, and to keep the blackboy if possible at his place until we came back.
After watching our blackboy well on his road and making sure that he had gone on the right track to the destination we had marked out for him, Logan and I began preparations for our expedition.

2-: From Chapter XVIII of Under the Southern Cross, a novel by N. V. Philpott, published in The Western Mail (Perth, Western Australia) of Saturday 17^th March 1894:

He went off toward the blacks’ camp when waiting any longer seemed hopeless. They all knew him as the “budgery ³ white fellow,” and greeted him with a cheerful “Hullo!”
He sought out the young gin of whom Lorne had been so fond.
“Kitty, will you take a letter for me?” he asked.
“Paper yabber?” she suggested.
“Yes, I want to take a paper yabber to Lorne.”
“Yoi!” with her soft gurgling laugh. “You write him now.”

³ The colloquial Australian adjective budgery, also budgeree, etc., means good, excellent.

3-: From the diary ⁴ of the Australian anthropologist and ethnologist Francis James Gillen (1855-1912)—source Spencer & Gillen: A journey through Aboriginal Australia:

October, 22^nd [1901]. Camp 63. Spencer and I spent the morning with our guns about the waterhole where we bagged three species of birds, two of which are new to us. This morning by way of experiment we started two Gnanji natives for Powell Creek with a note to Mr. Kell and wires for our wives. They are provided with as much tucker as they can carry comfortably and more baccy than they can smoke and I shall be surprised if they do not faithfully fulfil their contract although the distance 206 miles is much greater than I have ever known a blackfellow to carry a letter. The letter is carried securely tied in a cleft stick which is the usual method of conveying paper yabbers. The blacks regard messengers of this sort as sacred; they probably have an idea that if they interfered with a ‘paper yabber’ some evil magic would result.

⁴ In 1901-02, Francis James Gillen and the British-Australian evolutionary biologist, anthropologist and ethnologist Walter Baldwin Spencer (1860-1929) conducted a twelve-month anthropological expedition from Adelaide, in South Australia, to the Gulf of Carpentaria, on the northern coast of Australia.