If people can chew gum without making spectacles of themselves in the ways described above, I wouldn't forbid it. At the same time, it might seem distracting if any entire class were sitting there, chomping away. I doubt that this would happen, but I can imagine classes in which it might.
I, personally, don't chew gum, but I have no problem with students chewing it in class IF they chew and dispose of it properly. Sadly, when I taught middle school, the 6th and 7th graders absolutely could NOT refrain from (1) popping gum; (2) throwing gum and/or wrappers on the floor; (3) putting it in people's hair. I gave each class the option: dispose of the gum properly or lose the right to chew it. Rarely could a middle school class get more than a few weeks into the school year without me having to ban it. And every single year I had to deal with gum being deliberately put in girls' (and occasionally boys') hair; I cut it out more than once. Worse of all was me having to scrape it off my shoes and pants. Luckily, the problem was not as bad at the high school level, but even college students will spit their gum out on campus or in the parking lot, passing it on to others to pick up on the soles of their feet.
I don't think it's that big a deal. But it is essentially impossible to ensure that your students aren't sticking gum on the underside of desks, etc. So you have to depend on students to be mature which is not always the case. But I have never tried to enforce a no-gum rule and haven't had serious problems.
In college, where the rules regarding food are more lax, I knew one older student (late 60s, early 70s) who would routinely set up an entire meal on her desk during lectures. I never saw her called on it. That, in my mind, is more distracting than an entire room full of gum-chewers. As long as the teacher has no problem with the gum, and as #2 says, as long as the chewers practice reasonable silence with their chewing, I don't have a problem with it.
As a teacher, I have much better things to do with my time than fill out disciplinary forms on students who are chewing gum. For me, it comes down to a question of whether an activity disrupts the class or does damage. Students who snap, pop, or make other annoying noises with their gum are being rude and need to refrain. Students in my class are also expected to dispose of wrappers and used gum in the trash can, because it makes an unsanitary mess otherwise. I work with high school students, and I explain these thoughts to them at the outset and rarely have a problem, but much depends on the maturity level of the students.