Some more information:
When I configure cgminer to run both cards, it doesn't actually use the second one:
$ DISPLAY=:0 sudo aticonfig --adapter=all --odgc
Adapter 0 - AMD Radeon HD 7900 Series
Core (MHz) Memory (MHz)
Current Clocks : 1000 1250
Current Peak : 1000 1250
Configurable Peak Range : [300-1200] [150-1575]
GPU load : 99%
Adapter 1 - AMD Radeon HD 7900 Series
Core (MHz) Memory (MHz)
Current Clocks : 300 150
Current Peak : 1000 1250
Configurable Peak Range : [300-1200] [150-1575]
GPU load : 0%
The same occurs if I configure cgminer to directly use the second card (using -d 1).