Preheat grill. Brush the shrimp with olive oil and season with salt and pepper, to taste. Grill the shrimp until just cooked, about 2 to 3 minutes. Remove and let sit.
Arrange the arugula on a large platter. Top with the papaya and green onion. Whisk together the lemon and lime juice, soy sauce, jalapeno and olive oil until combined. Mound all of the shrimp in the center of the platter. Drizzle with the vinaigrette and sprinkle with peanuts and cilantro.