In a saucepan, bring the blackberries, 1/3 cup sugar, 1 1/2 teaspoons lemon juice and 1/4 cup water to a boil; let cool slightly. Transfer to a blender and puree. Strain through a fine-mesh strainer set over a bowl; discard the seeds. Refrigerate until cold. Divide the mixture among the molds and freeze until partially frozen, about 1 hour.
Meanwhile, in another saucepan, bring the raspberries, the remaining 1/3 cup sugar and 1 1/2 tsp. lemon juice, and 1/4 cup water to a boil over medium-high heat; let cool slightly. Transfer to a blender and puree. Strain through a fine-mesh strainer set over a bowl; discard the seeds. Refrigerate until cold.
Divide the raspberry mixture among the molds. Insert the sticks, piercing the blackberry layer, and freeze until solid, about 4 hours. Run warm water over the molds to remove the pops.
