In California, the onus is on you to fill this out. The dealer may do it as a courtesy, but they have no obligation to do so. It is your responsibility to make sure it happens.
Its a 2 minute activity, so yes, definitely go do it.
If it doesnt get done, your registration will come up for renewal. If you ignore it rather than make sure its dealt with, California will send you to collections for the past due registration you ignored.