Heat 1 tbsp oil in a large nonstick skillet over medium-high heat. Add shrimp and Italian seasoning; cook, stirring often, until the shrimp have become opaque, 3 to 5 minutes. Transfer the shrimp to a plate. Wipe the pan clean.
Add the remaining 1 tbsp oil to the pan; heat over medium heat. Add tomatoes; cook, stirring occasionally and pressing the tomatoes down lightly with tongs or a wooden spoon, until the tomatoes begin to break down and juices are released, about 5 minutes. Add basil and return the shrimp to the pan. Cook, stirring constantly, until the basil is wilted and the shrimp are warm, about 1 minute. Remove from heat; stir in pesto.
