There are professional and personal consequences you have to think about.
Professionally, the main thing is transfer of responsibilities. If you have no particular responsibilities, and are pretty much interchangeable with your colleagues, then there's nothing to do here. Some jobs are designed like that to allow for frequent staff turnover with minimal impact. Otherwise, this is your boss's responsibility to manage, but depending on your role you might be heavily involved in it or it might even be delegated to you. If you're telling people stuff about how to do your job when you're gone, you kind of have to let them know you're going in order to provide context for that.
From what you've said it seems like your boss is completely relaxed about this and foresees no problems. You might be inclined to follow that lead. However, maybe your boss is assuming you'll start telling people quite soon, and doesn't realise that your preference is not to do it at all. You should discuss this with your boss, because it's not clear whether "it is your personal business" is really intended to take precedence over "you need to do it". If you'd prefer for your boss (not you) to make an announcement at the latest possible time, then just ask.
Personally, there are likely to be people who'd prefer some sort of closure. If they know you're leaving they might stop by your desk to say goodbye, maybe exchange personal email addresses, or pay back the price of a coffee they borrowed from you 18 months ago when they forgot their wallet, never paid back, and would feel guilty about. Whatever's important to them. Furthermore, if you take a moment to say goodbye to them that's a social signal that they have meant something to you (perhaps not much, but more than nothing). So you can leave on better terms by giving people a short amount of time to do that -- perhaps a few days or even just a day, depending how easy it is to drop by and speak to you.
If your workplace is generally unpleasant, and you think it will be made even more so because your colleagues are horrible to people who are leaving, then they'll despise you whether you tell them or not, and perhaps you might as well sneak out. But if you're on good terms with them and just shy about big parties and rounds of applause, you can let them know you're going but privately ask your boss (who hopefully knows this about you already) to intercept anyone trying to organise farewell cakes/speeches/whatever. This will deal with most of the people who, as you've seen in other answers, will remain civil to you after you have left only if you go through certain routine pleasantries.
It's also worth bearing in mind that there are some (relatively few) workplaces where employees are walked out the door as soon as they hand in their notice. So the shock of a colleague leaving without notice clearly can be survived, but the difference from your POV between that situation and yours, is that in that situation everyone rightly puts it down to the employer's choice, and any ill-will falls on the employer. Whereas in your situation it's you who has chosen and they'll know it.
So no, you don't have to unless your boss instructs you to. Maybe not even then. Alternatives are that your boss does it instead of you, or that nobody does anything. Just be aware of the consequences when you decide.