Cut two pieces of cooking parchment to fit two 10- by 15-inch baking sheets. With a pencil, draw 10 evenly spaced 2-inch circles on each piece of parchment. Place each, marked side down, on a baking sheet.
In a large bowl or the bowl of a standing mixer, with a hand-held mixer or the standing mixer's whisk attachment, beat egg whites and cream of tartar at high speed until foamy. Add salt and beat until stiff peaks form, 2 to 4 minutes. Gradually add sugar, beating mixture until sugar has dissolved completely (rub a little between your fingers to check; if it feels gritty, sugar is not dissolved) and mixture is glossy, thick, and very billowy, 2 to 5 minutes. Beat in rum, then gently fold in coconut.
Mound mixture in 3- to 4-tablespoon portions in marked circles. Using a spoon, make a shallow depression in the center of each so it resembles a nest.
Bake cookies in a 250° regular oven (convection not recommended) until light golden, about 1 hour, switching pan positions halfway through baking. Turn oven off, leaving cookies inside, and let stand until dry and firm to the touch, about 2 hours longer. Remove from oven and let cool completely on sheets, then peel off parchment.
Set candies or jelly beans in each nest.
Note: Nutritional analysis is per unfilled cookie.