I have a 5DMIII with battery grip by canon.
By pressing the INFO button, I can only see the status of one battery. Am I not supposed to be able to see the status of both batteries in the grip?
It all depends on which Info screen you are talking about and what is currently displayed on the rear LCD when you press the INFO. button.
If you are shooting and you press the INFO. button to cycle to the Shooting Functions screen on the rear LCD, the value will be displayed as a combined one, as will the value displayed on the monochrome LCD on the top of the camera body.
If you press the MENU button and navigate to the Set-Up 3 tab (the 3rd wrench), and select Battery info., you should see separate values for each battery (Remaining cap., shutter count, and recharge performance) as well as a small symbol that shows in which side of the grip each battery is located. If you press the INFO. button while in this screen, you will see an additional screen with the location of each inserted battery as well as the serial number, last charged state, and the last date that the camera was powered on with each registered battery inserted in the camera or grip.
If you are only seeing one battery displayed in the Battery Info. screen, check to see that both batteries are securely seated inside the grip. I have experienced a few occasions when one battery was not firmly seated and didn't show up on the Battery Info. screen. Also check to see that the grip is squarely attached to the camera. There are over a dozen spring-loaded contact pins inside the camera's battery well that connect the electronics of the grip to the camera. One or more of these may not be making a good connection. One time this proved to be the reason the vertical shutter button would not function in one of my grips.
Additionally, if you are using any third party batteries they may or may not be chipped to correctly communicate with the camera. Normally when this is the case you will get a warning message from the camera when it is first powered up telling you the camera is having trouble communicating with the battery and asking if you wish to continue.
You can only see the detailed battery status under the Battery Info display in the menu. The LCD on top and Info screens show a combined status (average) for simplicity, though the camera will generally use power from both batteries roughly evenly.
The Battery Info screen (found under the third section of the Wrench menu) will display shutter counts for each (since last recharge) and the % of each. The info screen on that page will also display the last known percentages of any other batteries that have been used in the camera.
If you look on the menu under the "Wrench" Icon, third section, second choice -- "Battery info." Do you see both batteries? If so, all it good. It is my understanding that LCD and info screens only show a total battery power view -- meaning that when both batteries are ~1/2 discharged it should read 1/2.