Here's something approaching an answer, from
http://deepseachallenge.com/the-sub/systems-technology :
"The DEEPSEA CHALLENGER‘s pilot will be alone, but he’ll spend a lot of
the journey talking: He’ll narrate a “dive log” to two small cameras
(two are needed to create 3-D video) inside the sphere. He’ll also talk
to the surface through several systems—one with a 19-mile (30-kilometer)
range underwater. When necessary, he’ll be able to send text messages,
which are more efficient than voice communications. The submersible’s
mother ship, the Mermaid Sapphire, and the 30-foot (9-meter)
rigid-hulled inflatable Prime RHIB are each equipped with two
transducers—one on board and another suspended on a cable about 330 feet
(100 meters) below the water. The sub also has a backup acoustic
modem/transponder powered separately from the sub’s main battery. Once
the sub surfaces, it will also use a VHF marine radio to communicate
with the crew."
So apparently he has something resembling a very slow modem down there,
and the ship likely has an Internet connection via satellite or whatever.