Do doctors give anesthesia for ultrasonic scaling?

Yes sometime: Most of the time you don't need anesthetic with a regular cleaning with ultrasonic scaling if you have been going to the dentist frequently. You will need anesthesia if you need a deep cleaning because your gum is already inflamed and the scaler has to go deep under the gum to remove the deep calculus.

Usually: Most dentist will give anesthesia but usually for "deep cleanings" if you don't have gum disease and just need the anesthesia due to sensitivity you might want to try sensodyne toothpaste. You can also ask the dentist or hygienist to use topical anesthesia. If you still wanted the local anesthsia (lidocaine) you should be able to get it at a nominal cost.