Anyone getting an abortion (at least an invasive late term abortion that isn't just induced miscarriage) is going to have some form of anesthesia administered. Since they share a bloodstream, giving general anesthetic to the mother also anesthetizes the fetus. Whether local shares an effect or not I think depends on the type, but it is usually administered along with a sedative. It appears from this that most sedatives other than muscle relaxers will also sedate the fetus, but whether local anesthetic crosses the placenta depends on a lot of things: https://www.openanesthesia.org/placental_transfer_anesth_dru...
Here is a study of how effective various forms of anesthesia are for abortions, but it's not going to tell you the prevalence with which each form is actually used: https://www.ncbi.nlm.nih.gov/books/NBK561096/
Unfortunately, for various reasons, there is quite a bit of need for privacy on both the patients and the providers parts as the act of getting or performing an abortion will make you a target in a lot of places.
Here is a study of how effective various forms of anesthesia are for abortions, but it's not going to tell you the prevalence with which each form is actually used: https://www.ncbi.nlm.nih.gov/books/NBK561096/
Unfortunately, for various reasons, there is quite a bit of need for privacy on both the patients and the providers parts as the act of getting or performing an abortion will make you a target in a lot of places.