I have a 256 mb HD3850, I got it before the 512's were out. I can run COD4 on max settings @ 1440x900 and get around 40-50 fps on average (this varies depending on the map of course, but I never go below 30 fps).
Personally, I think if your running at anything above 1440x900, which includes 1280x1024, you should be getting 512. But for 1440x900 and below, 256 should be alright if you are on a budget.
Of course according to this review, it makes barely any noticeable difference adding 256 mb of vram to the 3850: