Mix the tomato paste, horseradish, olive oil, hot sauce, 1 1/2 teaspoons salt, and pepper to taste in a large bowl; add the shrimp and toss. Cover and refrigerate 30 minutes to 1 hour.
Preheat the broiler. Brush a rimmed baking sheet with olive oil. Spread out the shrimp on the baking sheet and broil until just cooked through, 4 to 8 minutes (no need to turn). Serve warm or at room temperature with lemon wedges.