Tubing manufacturers could supply stay ends in a variety of styles; e.g. from Tange's catalog:
A builder could form these from a square-cut stay end, but it's a fairly time-consuming process. Mass production frames generally use the treatments supplied by the tubing manufacturer. The radiused stay end treatment we used on the higher-end frames at Trek was formed from square-cut stays, filling the ends with brass when brazing the dropouts in place, then grinding the radius with a Dynafile: