When I was in high school, my Spanish teacher asked me in front of the whole class if it was possible for me to talk without moving my hands. I defiantly sat on my hands for the rest of the period...and couldn't finish a complete thought for the life of me. Every time I opened my mouth, my hands struggled to regain their freedom.
As I've gotten older, my gesturing isn't as expansive as once it was - I freely admit that it was dangerous to stand too close to me if I was in full story-telling mode when I was younger, particularly if you were holding a beverage - but it's still pretty well ingrained in the way I communicate. I still gesture more if I'm speaking Spanish instead of English, and those gestures tend to be more of a pantomime to help me find the right words. And I do still gesture while speaking to someone on the phone, or who could not otherwise see me.
Recently, a study came out that helped explain a little bit about why people gesture when they can't be seen. Basically, gesturing is part of learning a language - even people who have never seen another person gesturing as they speak will have very similar hand motions as they speak. It's a fascinating way of looking at language and how it ties back into the rest of the body, above and beyond just speaking and listening.
Even with the basic gestures being similar among speakers of the same language, however, everyone does things a little differently. Some of us are more enthusiastic and descriptive with the gestures, while others may keep their hands contained and their motions minimal. It changes how other people perceive them, and it can change with the character's mood - I know that if I'm not doing well emotionally, my hands hardly move at all when I speak.
Adding another way to illustrate a character's method of communication can help a character's development gain some depth and realism. Also, it can just be fun to see how much a character will flail when you start to put them in frustrating situations.