Thread scallops, through width of the rounds and touching, onto a thin metal skewer. Push a second metal skewer through scallops, parallel to and about 1/2 inch from the first. Thread shrimp, touching, onto thin metal skewers.
Rinse and drain radicchio, Belgian endive, and arugula. Separate leaves from radicchio and endive heads. Discard tough arugula stems.
In a large bowl, mix oil, vinegar, and chives.
Lay scallops and shrimp on a barbecue grill over a solid bed of hot coals or high heat on a gas grill (you can hold hand at grill level only 2 to 3 seconds); close lid on gas grill. Turn as needed until scallops are lightly browned and shrimp are pink, and both are opaque but moist-looking in center of thickest part (cut to test), 5 to 7 minutes for scallops, about 5 minutes for shrimp.
Add radicchio, arugula, and endive to dressing; mix. Lift salad, draining briefly, onto plates, dividing equally. Push scallops and shrimp from skewers into bowl. Mix, then spoon onto salads. Add salt and pepper to taste.