For people to travel to Barsoom (Mars) or to Earth (Jasoom) or any other planet with that specific device they must say the chant (ohm octei etc. and planet name) themselves, while holding the device, as we have seen in the movie. But in the end the therm sends John to Earth by saying the chant from his own mouth. How can this be possible?
In the books the talisman doesn't exist, so we can't refer there for more information.
But, given the scene where they visit the "map room" they discover that they can control and manipulate the devices without any access codes or other form of authorization. Same holds true for the weapon given to Sab Than (he could use the device immediately after receiving it, leaning to the idea that anyone can use any of the devices at any time.
Going along these lines, the chant would need only be spoken to activate the device, then the device (and any people attached) would then be transported to the targeted location. Given that no other device shown has any restrictions, it would stand to reason that the talisman would not either.
Also, note, that the Thern that Carter kills in the cave speaks the incantation that transports him up to Mars in the first place.
EDIT: My bad, still early. Refined point and added more info.
My guess: "Okh Ohem Oktei Oeeis* (destination)" is ONE command, a Thern would know others, and we never hear all the syllables when Matai Sheng sends him back. There might be a "send myself there" command, and a "send THIS guy there" command.
*It strikes me as cool to spell it assonantly, so I do. But who knows for sure how to spell it. Where's that darn Disney screen play, when you need it.
Edit: correcting myself already, huh. Spun the tape back, and it seems we do hear Matai Sheng do the whole command when sending Carter back! He just does it very fluidly, four words become one, sounds like "okomoktewis Jasoom." There would almost certainly still be a lot of variability in doing it, available to the operator.