Add the paprika, minced garlic, Italian seasoning, lemon juice, olive oil, black pepper, dried basil, and brown sugar to a large mixing bowl.
Whisk thoroughly until the brown sugar begins to dissolve and the marinade is evenly combined.
Add the peeled and deveined shrimp. Toss gently until every shrimp is well coated with the garlic and herb marinade.
Cover the bowl and refrigerate for at least 2 hours. Turn or stir the shrimp once during marinating to redistribute the seasonings.
Preheat an outdoor grill to medium-high heat. Clean and lightly oil the grill grates to help prevent sticking.
Remove the shrimp from the marinade and allow the excess liquid to drain away. Discard the marinade that has been in contact with raw shrimp.
Arrange the Grilled Garlic and Herb Shrimp on the hot grill in a single layer. Skewers can also be used to make turning the shrimp easier.
Grill for approximately 2–3 minutes on the first side.
Turn the shrimp and grill for another 2–3 minutes, until they are opaque, lightly charred around the edges, and cooked through.
Remove from the grill immediately and serve warm with fresh lemon wedges, if desired.