Are you sure that is *the* Coptic tradition and not just one way it's done in the churches?
The tradition I'm familiar with is an old man shoves a cloth at you, another rips it from your hands moments after the priest places the Body in your mouth and shoves you out to keep the line moving, and then a third scolds you for not having the cloth held to your mouth until you finished swallowing...A very spiritual exercise devised by the holy fathers to instruct us in patience and long-suffering.
In all seriousness though, this tradition is interesting because of it's origin. In the first centuries in the Coptic Church, Communion was received in the hand. The cloths originated out of respect, with the communicant placing the cloth over the hand (with the left hand cupped behind the right hand), the priest placing the Body on the cloth in the hand, and the Communicant reverently lifted the Body to their mouth. People were also advised to place their finger on their lips while still moist from receiving the Blood and anoint the eyes with it. Practices that are certainly not allowed today.
Today the priest places the Body in the mouth (obviously the change came about do to abuses), and the person places the cloth over the mouth. The story has originated that this is to prevent any particle from falling. But this is just a story. It is quite unnecessary to hold a cloth over one's mouth to keep food from falling out, we do it at every meal. In fact, if we really though the cloth was being used to hold the Body in we would treat it with much more respect and have the priest wash it and dispose properly of the water in a garden or by drinking since it would have touched the Body. No, the cloth is just a very long-living hold-over from the more ancient practice, an echo of receiving Communion in the had that has lasted for all these centuries.
When the priests commune, they do not hold cloths over their mouth (since they touch the Body with their hand, and don't need to place a cloth over it to receive in the hand). It is not necessary to use a cloth. It is not required in any canon or rubric. It is just a pious custom, a I wish it were regarded as such rather than being regarded as a core tenant of Coptic Tradition.