Put yourself in the shoes of Zecharias's peers who were, rightly, concerned about his long delay in the Holy Place. Perhaps God had struck him dead! Perhaps he had had some sort of physical symptom that incapacitated him for a short time! They just didn't know.
When Zecharias does appear, then, the first thing they ask him is
Hey, Zach. What up, man? Did you have a heart attack or something? We were worried about you, man!
Now, what is the first thing Zechariah tries to do? Yes, he tries to speak but is unable to speak. His peers, obviously confused, are NOT aware that he is ONLY mute and NOT mute AND deaf, so they just automatically start signing and gesticulating wildly in order to communicate with their good friend Zach. I imagine Zach got them eventually to realize he was mute, only, and not deaf, so they probably worked things out with ink and a piece of papyrus paper, or with a tablet, just as Zecharias did nine months later with perhaps a different group of people, the composition of which is not certain.
This may seem cruel, so forgive me, but sometimes people get confused when they're introduced to a blind person, and they start talking as if the blind person can't hear them, which may instigate a curt "Hey, I'm blind, not deaf" from the sightless person.
By the way, one commentator I read (viz., Constable), suggests Zecharias was both mute and deaf. I think I disagree with Constable. First, Gabriel did not say Zacharias's punishment would include deafness, though as Constable says, "The Greek word used to describe [Zach's] condition, kophos, can mean deaf as well as dumb (cf. 7:22)." Second, how did Zach know there was a movement afoot at John's circumcision to name the baby Zecharias and not John? Perhaps someone wrote a message to Zach on a tablet to that effect (and that is certainly possible), but just as likely, I suggest, is that Zach heard what the people were planning to name his son, and he would have no part in it. He therefore called for a tablet and wrote, "His name is John."