The only power tool I have is a drill. I have a set of bits, too.
How can I make the edge of a wooden dowel rounded off inwardly, so I can place on another dowel?
This is the shape I’m looking for:
This is the effect I’m trying to achieve:
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You can use your drill. You'll need a bit whose size is the same as the diameter of the inner curve you want to cut. Clamp your dowel stock firmly, position the drill so the outer edge of the bit is at the spot you want the curve to be, and drill. Clean up the cut, if needed, with some sandpaper wrapped around another dowel.
As @Graphus mentioned in a comment, the best way to secure the dowel is in a v-notch, which you can make easily with scrap wood.
If you don't have a saw, two scrap pieces of wood with a small gap between them will also work.
This will be easier on a drill press, of course, and may be easier with a forstner bit. You'll get better results if you cut to length after.
Nick Ferry has a YouTube video showing this process, which he used to make some drawer pulls.