No, you don’t.
There are people who use CSP with a mouse. I wouldn’t recommend that, but it’s been known to happen.
If you’re just starting, I’d recommend getting a (relatively) cheap tablet to draw on. Wacom’s base model is the Intuos and can be had for under $100. If you go with the Intuos Comic it even comes with a limited time (two years) version of Clip Studio Paint.
If you want to go cheaper, Monoprice offers a couple off-brand options for you.
I used a Wacom tablet for a decade before the stylus that went with it died. Those stylii are impossible to replace now, which is why I moved to a Monoprice special (which is the same as the Huion H610, basically) after that. They all worked just fine for me, though the Wacom felt heavier and less flimsy. Those Monoprice/Huion stylii feel fragile in comparison.
It will take some getting used to to draw on a tablet while looking at the computer screen. Ultimately, if you get far enough, you might find that hindering your skills. At that point, you should look into a digital pen display. The Wacom Cintiq rules the roost in that market, though Amazon and Monoprice have far cheaper alternatives that are pretty good, too. Check out Frenden’s reviews of those. I use a Yiynova 20″ display, for example. (Here’s a good review of the Yiynova 19″ that I found useful in making my decision.)
If you’re just starting out, your best bet is likely to buy the Wacom Intuos Comic tablet. It’s $100 at Amazon, and comes with a two year license for Clip Studio Paint. Get a good feel for it. Use it for awhile before you decide if you want to step up to a display type device or just a larger tablet.
I know it seems like a lot to spend, but if you are serious about drawing digitally (even is just as a hobby), it’s an investment that is well worth the money. Unlike other hobbies, you won’t need to keep spending money to use it. Buy once, use for years. Brilliant!
Eventually, if you go far enough, you’ll want a digital pen display like a Cintiq or a Yiynova. But you don’t need to start there. See if the tablet improves your art first. When that starts to become frustrating for you, then start saving your pennies for something more direct.
What device are you using? Do you have a fancy touch screen Cintiq? A generations old Wacom tablet? A third party pen display? Leave a comment below. Let’s see what we’re all using here.