Bring a large pot of water to a boil. Add soba noodles; cook in the boiling water until tender, about 6 minutes. Drain and transfer to a large bowl.
Pour 1/2 cup chile dressing over noodles; mix thoroughly. Cover bowl with plastic wrap and place in the refrigerator.
Combine green papaya, green onions, string beans, cherry tomatoes, 1 teaspoon black sesame seeds, cilantro, and red pepper flakes in a bowl. Pour in remaining 1/2 cup chile dressing; mix thoroughly. Refrigerate until flavors combine, 15 to 20 minutes.
Divide noodles among serving plates. Spoon papaya mixture on top. Garnish with remaining 1 teaspoon sesame seeds and sriracha sauce.