This site is marked on the 1:25000 OS map as 'standings stones', but there is definitely only the one here (as confirmed by the RCAHMS notes). It stands next to a dry stane dyke at the entrance to a ploughed field. It's aligned NW/SE and there is a very curious 'seat'-like shape to the stone- whether natural or sculptured it's difficult to say, but there's a seat part and the bulk of the stone forms the back part! The stone is 1.47 m high by 1.2 m wide and around 36 cm thick. The E side has the 'seat'- it's 42 cm wide and 1 m long. Nice spot for a sit down!
Now I'm not 100% certain this is the right stone. But it could well be. The article mentions Cowdenknowes and Smailholme, and Gordon, all of which are near. Also the cottage - which although isn't there now, is mentioned on the RCAHMs record as having been demolished .
On a knoll by the side of [the] road there was an old thatched cottage, with an immense upright block of stone at the end of it. The place was called 'Standing Stone,' and there was a popular rhyme attached, which used regularly to afford us matter for the most serious inquiry, whether superstitious, mythological, or historical; shedding also a mysterious interest on the house itself and its inhabitants.
The doggerel couplet involved a favourite quirk with the vulgar of most rural districts, though somehow or other it always seemed to have in this case an unusually imposing effect--
'When Stannin'-Stane hears the cock craw,
It wheels about, and faces Gordon Law.'
He then goes on (at great length) to describe a boyhood incident when there was a 'the most awful thunder-storm I ever witnessed' and he and his friend were terrified by the stone's 'black shape as silent as death' waiting to act as their gravestone. 'Drenched we were to the skin, yet couldn't think of going up to ask shelter.'
From Chamber's Edinburgh Journal of Nov. 11th, 1848.
An entry from Ancient Stones, an online database that covers most of the standing stones, stone circles and other stones found in South East Scotland. Each entry includes details, directions, photograph, folklore, parking and field notes on each location.