The short answer is: no.
Of course you don’t. The only thing you need is your body – duh. That said, if you’re preparing for a race and care about your race time, or if you’re trying to set a new PR: then yes, definitely. I got my first Garmin watch a bit prematurely in a way: I wasn’t running very often, and though I had signed up for a marathon, I wasn’t particularly concerned about my time. The marathon was a fun way to push myself. I was curious about my heart rate though and figured that if I did wind up falling in love with running, as I sensed might be the case, that a watch would inevitably come in handy.
Here’s what I noticed: the minute I received the watch I had to go for a run. I wanted to test it out and see if I liked it. And I did! Eventually, I came to realize that having made the investment, I felt obliged to use it, and that it was pretty cool to see how my heart rate would fluctuate, what my cadence was, and how fun it was when I sensed a buzz on my wrist because I’d knocked down another mile.
For me, new gear is usually helpful, but the effect it has is sometimes more important: it makes me want to use it, and hence look forward to a run I might not otherwise. If you’re not using one now, you may find that once you do, the data and information you’ll learn about yourself will be interesting, maybe eye-opening, and probably useful. It also doesn’t hurt that the one I got I found to be puh-reeety pretty.