John Singer Sargent, "The Daughters of Edward Darley Boit," oil
To me the best and most straight-forward explanation for the seeming randomness of the girls positions comes from MFA curator Erica Hirshler. In her book "Sargent's Daughters, the biography of a painting", she explains that Edward Boit being an artist himself, and understanding the amazing talent that Sargent was, gave him a tremendous amount of artistic freedom to do whatever he liked with his subjects and their surroundings.
Another interesting point I’ve read concerning "The Daughters of Edward Darley Boit" is the statement Sargent seems to be making about women coming of age. He presents us with various stages of life from the innocence of youth represented by the doll, and the youngest girl holding it - both of which are in full light - to the more mysterious, inwardness, and awkwardness of young adulthood, represented by the oldest daughters, one of whom is not even facing us from the shadows.
None of these points are 100% certain mind you, and I like that fact because it only ads to the mystique of the piece. One thing is for certain, few other painters in the history of art have had the ability to observe and use so many colors to depict the color white - something Sargent gave himself the opportunity to do in this and so many other paintings (as shown below).
John Singer Sargent, “An Artist in his Studio,” oil