Noise varies with paddle, for me anyway.
I was messing around with three paddles today in my Curtis Ladybug and they varied in noise / vibration with in-water recoveries. Listed with the quietest first:
1. 1988 era Grey Owl Freestyle, almost completely silent.
2. Camp beavertail (about 5 years old) bought on ebay from Foxworx, most similar to the Foxforx Foxtail in their current line up. Blade dimensions: 6.0" x 27" (longer and narrower than the current Foxtail).
Edit: It was probably a special order or experimental model (I just talked with Foxworx). Foxworx would make me a custom size to my preferred blade dimensions for a little more money.
3. Grey Owl Tripper (current model, I think).
My expectation is that different specimens of the same models would have either more or less noise depending on the finished shape and smoothness of the surface.
I suspect that they'll all get a little quieter as my technique improves.
I'd never tried the indian stroke until today and really enjoyed it while exploring a lake with lots of nooks and crannies. It felt most natural with the Foxworx paddle and then the Grey Owl Tripper. The Foxworx has a thicker shaft and larger grip. It took a few minutes to adjust to the different shaft and grip size when switching between those two paddles.
Thanks for all of the great descriptions and discussion.