I planted four patches of Scottish moss last May in an area, where native moss grew, moist and shady. By late July the patches had grown thin and scraggily and appeared on their way to oblivion. Trying this site or one like it for advice came up empty. So I transplanted it into a bright sunny spot in July/August. By fall the patches had not only bloomed and thickened, but also spread to about twice their size at the time of transplanting. Ooops, correction three of the four had done so. The fourth was apparently to far gone already and dried up. We live in a very dry and hot region. I am sure the temperature approached or maybe even crossed 100^ F after transplanting. So I would vote for sunny conditions, despite some evidence to the contrary in this thread.
Next spring I will look for one or two different shades of Scottish moss and plant them also in full sun.