Pat duck dry, then prick skin of duck all over with tip of a sharp knife.
Heat oil in a wide 6-quart heavy pot over moderately high heat, then brown duck in 3 batches, skin side down, turning over once, 8 to 10 minutes per batch.
Transfer duck to a bowl and pour off and discard all but 1/4 cup fat from pot.
Reduce heat to moderately low, then add flour to fat in pot. Cook roux, stirring constantly with a wooden spatula or spoon, until well-browned (a shade darker than peanut butter), about 20 minutes. Add onions, celery, bell peppers, bay leaves, and salt and cook over moderately low heat, stirring occasionally, until vegetables are crisp-tender, 6 to 10 minutes. Add broth, water, and duck with any juices accumulated in bowl and bring to a boil, then reduce heat and simmer, uncovered, until duck is tender, 1 1/4 to 1 1/2 hours.
Remove gumbo from heat, then transfer duck to a cutting board with a slotted spoon and shred meat into large pieces, discarding bones and skin. Skim fat from surface of gumbo, then return duck to gumbo. Bring to a boil, then reduce to a simmer and stir in shrimp, scallions, and cayenne. Simmer gumbo until shrimp is just cooked through, about 2 minutes. Discard bay leaves. Serve gumbo over white rice.
Cooks' note: Gumbo, without shrimp, scallions, and cayenne, can be made 3 days ahead and cooled completely, uncovered, then chilled, covered. Bring to a boil, then reduce to a simmer and add shrimp, scallions, and cayenne and simmer until shrimp are just cooked through, about 2 minutes.