Place red potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender and the peels begin to crack, about 20 minutes. Drain and let potatoes cool. Peel cooled potatoes and cut into 1/8-inch dice.
Place potatoes, shrimp, onions, hard-cooked eggs, sweet pickles, and mayonnaise into a large salad bowl and stir to combine. Season with salt and black pepper; sprinkle top of salad with paprika. Chill salad at least 3 hours before serving.