Peel shrimp, keeping the tails intact. Place in a bowl with 4 tablespoon of the rum, lemon or lime juice, oil, garlic, red-pepper flakes and salt; toss to coat thoroughly. Cover and marinade in the refrigerator for 30 to 60 minutes. Meanwhile, prepare a charcoal fire or preheat a gas grill.
Combine brown sugar, vinegar and the remaining 2 tablespoon rum. Thread shrimp onto skewers and lightly brush with brown sugar glaze. Grill over a medium hot fire, brushing with additional glaze, until shrimp are opaque, about 3 minutes per side.