I see nothing on Willie Watson but there appears to be a certain amount of surviving correspondence on a number of other early professionals mentioned here on this thread including the Foulis brothers.
It appears this correspondence is in and around Shinnecock (Shinnecock's archives) and it includes correspondence from 1893 to 1896 between Samuel Parrish, Thomas Barber and Edward Mead of Shinnecock on this side involving the selections of professionals from abroad to America. It includes the mention of James and Robert Foulis, Andrew Kirkcaldy, Willie Park Jr and John D. Dunn.
The correspondence from the other side includes some of those mentioned as well as Old Tom Morris and particularly a man by the name of Henry White. Henry White, also known as the Marquis of Granby, was with the American Legation in London but he was also the editor of the UK's Golf magazine.
In other words, it appears when the likes of Parrish, Barber and Mead wanted information abroad on golf professionals around 1893-1896 their first step was to contact Henry White of the American Legation in London.
It also appears from the above that James Foulis went to the Chicago GC in 1893 and that his brother Robert followed him around this time and went to Onwentsia GC. Robert Foulis was recommended by White for Shinnecock but obviously he went to Chicago and Onwentsia. In 1895 Andrew Kirkcaldy was recommended to join Willie Dunn at Shinnecock but he declined citing a match abroad against J.H. Taylor. It also appears Shinnecock did not have a professional in 1893 or since the departure of Willie Davis in 1892 until Willie Dunn came to the club in 1894. In 1896 Dunn departed Shinnecock to go to Ardsley Casino and Shinnecock's next professional in 1896 was a man by the name of R.B. Wilson.