If the defendant is in default, you have no obligation to mail anything further to the defendant. However, you may wish to start the ball rolling regarding time to appeal and/or to set aside the judgment. If so, notice is an important part of that equation. If I were representing you we would discuss the matter first but I would probably advise giving notice of entry of judgment. Remember, as a party, you cannot serve documents by mail yourself. A disinterested adult must serve. Good luck.
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Mr. Daymude's advice is spot on. You do not have to serve a copy of the proposed judgment on the debtor if his default has been entered. However as Mr. Daymude advises, you might want to do so as soon as possible after entry of judgment in order to start the clock running on the debtor's right to have the default set aside under CCP §473.
I serve notice of entry of judgment as a matter of course, whether the judgment is by default or after trial. A little trick I then use if the judgment is by default is to let the judgment "season" for six months after service of notice of entry. In other words, I advise my client that we should make no collection efforts until termination of the six-month period to avoid a motion to set aside the default. The judgment collects interest at 10% per annum, and that ain't bad. The only caveat to this advice is if you think the debtor is sequestering his money or assets. Then you have to move with alacrity.