For the cocktail sauce ... and Worcestershire sauce; pulse to combine. Scrape into a serving dish and refrigerate at least 30 minutes or up to 4 hours. For the shrimp: Preheat the oven to 425˚.
Set aside to cool. For the sauce, combine the chili sauce, ketchup, horseradish, lemon juice, Worcestershire sauce, and hot sauce. Serve as a dip with the shrimp.