The operator needs to sleep. The issue is that because of the pressures of research, video-blogging and the additional tutoring he's getting on Na'vi society (as well as the time he's spending awake in the avatar body), Jake isn't being allowed to sleep in either body.
Simply put, when the operator is controlling one of the bodies, either their own or the Avatar body, they're awake. When they're asleep in either body, they're asleep.
The official Avatar : Pandorapedia states that when the consciousness is elsewhere, the vacant body is in a sleep-like state. Note that while the body is catatonic, the mental functions are completely intact. The script states that it appears that the operator's facial expression is similar to "a dreamer in REM sleep..."
While the human controller remains in a sleep-like state in a psionic
link unit, his or her personality inhabits and completely controls a
custom-made Na’vi body.
Within the script itself, we see the consequences of this sleep deprivation on Jake:
Jake looks like he’s about to collapse. He’s gaunt, eyes deeply
shadowed, hands shaking.
TRUDY : You need to get some rack.
Jake grabs a packet of freeze-dried crystals and pours them directly
into his mouth, chewing them up.
JAKE : Gonna have to settle for coffee.