I sewed my panels with one straight side simply because the curved side is a little trickier to get looking smooth. But both sides could be finished with a curved bottom to resemble a feather, as you say. I assembled the panels in random fashion, not putting much importance on which side was towards the center. Try pinning the whole thing together on a band and see which way you like it, before stitching. I'm lucky enough to have an industrial Juki machine available to me, which has a tiny feed dog and foot that edge-stitches fine fabric like silk beautifully.
Some sergers have a setting for a tiny overlocked thread edge.
With a regular home sewing machine it might be more difficult. Buy a small sample of whatever fabric you want to use, and try making some narrow double-turned hems , including curves. I do these in one operation, but doing it in two steps is fine too. If it's making you tear your hair out, use an easier fabric. Experiment with any fabric you like.