Jumping in a bit late on this question, but if you’re moving to a very similar position in a similar industry, it’s likely that word will get around even if you don’t tell them. I don’t know your industry or the Bay Area, but even in large cities, many industries are pretty small and you old boss may know your new boss or someone else in the new organization. The people at New Company know where you came from so they easily could say to their buddy at Old Company, “So Red jumped ship to come work for us. Guess you weren’t paying enough! Hehehe.” Or whatever. And suddenly Old Boss knows where you went, which could be uncomfortable if you are friends with Old Boss and/or will see them at professional conventions or something similar.