On my machine (AMD A8-3500M/Radeon 6620G) using the r600 Gallium3D driver as shipped in Ubuntu 14.04 (SO, Mesa 10.1 + kernel 3.13) causes the Steam Overlay to simply not run.
It gives me errors about gameoverlayrenderer.so not being the right ELF class and Shift+TAB just starts cycling backwards through the game menu. Games themselves all run on r600g, and surprisingly well at that, I just can't use the overlay.
Using Catalyst results in the overlay working like it should, although my machine then becomes a little less stable. I tried using the updated Mesa/DRM from oibaf's PPA, but it didn't help. Tried using the latest beta of the Steam client, also no change. Tried purging and reinstalling Steam, also no change.