To those of us that have seen an example of this scarce penny, it is surprising that some people misidentify the coin. But I have seen 1919 pennies mistakenly identified as double dots by collectors and in stockbooks at shows, in dealers' shops and in internet auctions. The latter of course does not surprise, but the other instances do.
The 1919 double dot penny is quite different in appearance to its 1920 counterpart in that the dots on the 1920 double dot are smaller and better centred than those of the 1919.
There is a 'damaged die variety' that has sometimes been mistaken for the 1919 double dot. It is best summed up by this quote from Mike Locke:
1919 #3C as #3A but with large dots above top scroll and below bottom scroll. A small die lump appears at the top of "Australia". The upper dot sits in a well, giving the appearance that it was added post strike. This is a popular, scarce and pricey variety. Don't be fooled by the variety listed under cracks and chips.
1919 Cracks and chips Variety 2B, crack through date, tiny rust pit above top scroll simulates the two dot coin.
Unfortunately I do not have a specimen of Mike Locke's 1919 2B to show here, but I have seen them. From what I can remember, the 'dot' is tiny, to the right of centre and under magnification does not look anything like the result of a dot punched into a die.
There are a couple of other things to look for on a genuine 1919 double dot, the die damage at the top of the A of AUSTRALIA and the fact that the dot below the bottom scroll is positioned to the left of centre.
From searching through bulk penny lots, I believe the number of 1919 double dot pennies to be similar to the number of 1920 dot above top scroll pennies - an estimated mintage of 20,000 to at most 40,000.
The 1919 double dot is almost always found only in lower grade; fine is extremely scarce, VF or better rare. I've heard of only two specimens that possess lustre (but have never seen any). This is the best I've managed to obtain for myself.
Notice that the reverse of even a higher grade 1919 double dot has the same wishy-washy look that the low grade ones have. I suspect that the flat-top appearance of the inner beads is due to the die itself rather than wear, although I'd have to see one of those with lustre to be certain.