The Tech Report posted a review on Diamond's Radeon HD 3870 1GB Video Card. Here's a taster:

Raising the Radeon HD 3870's memory size from 512MB to 1GB doesn't seem to improve performance measurably in any of the games we tested, even at a monster 2560x1600 resolution. As a result, the Radeon HD 3870 1GB delivers 3D gaming performance that's usually lower than the GeForce 9600 GT's.

This outcome isn't really surprising to us. Radeon HD 3870 cards seem to get along fine with 512MB of RAM, even in extreme circumstances. We saw strong performance scaling from the CrossFire X multi-GPU scheme, which imposes some memory overhead, in Crysis with three and four Radeon HD 3870 GPUs, each with 512MB. In the same situation, multiple G92 GPUs with 512MB ran into an obvious performance wall, while their 768MB cousins did not. AMD clearly has its GPU memory management mojo working better than Nvidia's, so going beyond 512MB isn't yet necessary.