As a data scientist living in the South, I have no problem relocating to San Francisco or San Jose, but how do I signal this without letting my existing employer know? Changing my LinkedIn location from <tiny city> to San Francisco would raise some uncomfortable questions.
Maybe it's ok to let your employer know? It's not as if the desire to relocate could be solved by your current company if all of their operations are local. I suppose it depends on your manager, but hopefully if you said "Boss, I want to move to San Francisco. The growth opportunity there is too great to ignore." they might be supportive. Obviously if you think that conversation could go very badly then you might have to be a bit more low-key. Maybe try to reach out to a recruiting firm in San Francisco instead of changing your LinkedIn.
Disable the notification when you change your profile. Noone is gonna notice that you changed.