A character gains the stone's benefit while it's in their possession.
The description of the School of Transmutation wizard feature Transmuter's Stone says, in part (PHB, p. 119; emphasis mine):
Starting at 6th level, you can spend 8 hours creating a transmuter’s stone that stores transmutation magic. You can benefit from the stone yourself or give it to another creature. A creature gains a benefit of your choice as long as the stone is in the creature’s possession. [...]
As long as the stone is on a character's person, that character gains the benefit from the stone that you chose when you created the stone (darkvision, a speed increase, proficiency in Con saves, or resistance to one of the listed damage types). This applies whether the character is holding it, wearing it in a necklace, has it in a backpack, or however else they might have it, as long as it's on their possession.
(If you have the transmuter's stone yourself, you can also change the benefit of the stone when you cast a transmutation spell of 1st level or higher as long as it's in your possession.)