Essays
Three essays of the two denominations were prepared by William Gullick (the newly appointed Government Printer for NSW). Details of these are obscure. Richard Peck reports they are held in the NSW Archives. There is one poor reproduction on the cover of Sydney Views for May 1888.
Proofs of the 1d (1/) stamp.
Proofs of the 1d (1/) design were printed in the issued colours on thin, unwatermarked card.
They were perforated 11 x 11.
The number of proofs printed is unknown but it likely to be only one sheet of 30. Very few proofs are recorded as still existing (probably less that 10) and it may be that there are only the three described below.
It had been generally accepted that all the proofs were overprinted SPECIMEN but the second example shows that is not correct.
Proof overprinted SPECIMEN.
Five unclear stars.
Perforations in lower right corner fairly straight at a 45 degreee angle.
Status April 1982 Lot 182.
Status July 2002 Lot 1241.
Millennium Auctions Nov. 2012 Sale 54, Lot.809.

Proof
NOT overprinted SPECIMEN with slight damage.
Top right star very faint.
Status July 2002 Lot 1243.
Millennium Auctions October 2002, Lot 207. 
Proof overprinted SPECIMEN.
5 clear stars.
Perforations in lower right corner flattened
and several along the right side are comparatively longer.
Prestige Auctions, May 2012 Lot 1134.


The 1d (1/) proofs differ from the final designs in a few minor details  for example the proofs have:
 only three or four poorly formed stars for the Southern Cross;
 more brown shading dots on the bodies of the angel and the sick person.

Proofs of the 2½d (2/6) stamp.
Proofs of the 2½ (2/6) stamp are equally rare. It is considered on the evidence below that five proofs exist. Some sources claim only three proofs exist.
The respective proofs can be distinguished by the position of the "SPECIMEN" overprint relative to the gold decoration before JUBILEE.
From their
position, the overprint must have been applied separately from a single handstamp and not from a forme.
Proof overprinted SPECIMEN.
 "S" is the furthest to the left of all examples;
 "m" is well clear to the left of the gold decoration;
 the "e" of "en" clearly overlaps on both sides of the gold decoration;
 the left vertical of the "n" is clear to the left and below the upper protrusion of the decoration.
Millennium Auctions:
Rarities October 2002 Lot 208;
Nov. 2012 Sale 54, Lot 809.

Proof overprinted SPECIMEN.
 "m" is well clear to the left of the gold decoration;
 the "e" of "en" is coincident with the left side of the gold decoration;
 the left vertical of the "n" emerges from the left of the upper protrusion of the decoration.
Corinphila, May 2018, Lot 62. 
Proof overprinted SPECIMEN with some damage.
 "m" is just clear to the left of the gold decoration;
 "en" of the overprint overlaps the decoration;
 the "n" is just clear of the lower protrusion of the gold decoration.
Stanley Gibbons September 2001 Lot 1312.
Prestige Philately Aug. 2012, Lot 259.

Proof overprinted SPECIMEN with some slight damage on the corners.

Detail of Proof overprinted SPECIMEN 
 "S" is the furthest to the right of all examples;
 right leg of "m" is just covered by the left of the gold decoration;
 "en" of the overprint just touches the decoration;
 the "n" is just clear of the lower protrusion of the gold decoration.
Status Auctions, July 2002 Lot 1242.
Corinphila, June 2023, Lot 20085. 

Detail of the above Proof overprinted SPECIMEN.
 "m" is just clear to the left of the gold decoration;
 "en" of the overprint overlaps the gold decoration
from about the centre;
 the "n" overlaps the lower protrusion of the decoration.
Status Auctions, July 2002, Lot 1241.
Prestige Philately, May 2013, Lot 1134.
Corinphila, November 2022, Lot 10077. 