Whisk the paprika, garlic, Italian seasoning, lemon juice, olive oil, pepper, basil, and brown sugar together in a bowl until thoroughly blended. Stir in the shrimp, and toss to evenly coat with the marinade. Cover and refrigerate at least 2 hours, turning once.
Preheat an outdoor grill for medium-high heat. Lightly oil grill grate, and place about 4 inches from heat source.
Remove shrimp from marinade, drain excess, and discard marinade.
Place shrimp on preheated grill and cook, turning once, until opaque in the center, 5 to 6 minutes. Serve immediately.
