Thaw raspberries, if frozen; do not drain. Place sugar in a shallow bowl. Rub lime wedge around the rims of two 6- to 8-ounce cocktail glasses. Dip rims into sugar to coat; set aside.
If desired, set aside several raspberries for garnish. Place the remaining raspberries in a food processor or blender. Cover and process or blend until smooth. Strain berries through a fine-mesh sieve; discard seeds. Set aside.
Divide the snipped mint between the prepared cocktail glasses. Fill glasses three-fourths full with ice cubes. In a cocktail shaker, combine rum and the strained raspberries. Add ice cubes to fill shaker almost full; shake until very cold. Strain into the ice-filled glasses.
Slowly pour carbonated water into glasses. If desired, garnish with reserved raspberries and mint sprigs.