I am not aware of a stethoscope-stealing epidemic. What does occasionally happen is that you put it down somewhere, forget, and then someone else either tidies it up or "borrows" it and puts it down somewhere else. If your name is in it then the chances of it eventually making its way back to you are significantly higher. I've had two go missing for months at a time and then got texts from colleagues saying "did you know your steth is hanging up in the ICU handover room/theatre 5?" It's also much easier to send out a group text/email looking for it if it has some identifying features.
So yeah, get it engraved for identification purposes but don't get too stressed about stethoscope thieves.