OMG, I'm not alone.... Sorry I don't have a solution... I just put together a new system (including new video card and monitor). My video card is XFX Radeon HD 5770, and my monitor is Samsung P2370HD (23-inch, Full 1080p (1920x1080)). I'm having the same issue.
With HDMI connection I have a black border around the entire screen, and poor image quality (slight distortion). It's very easy to notice the quality because in my troubleshooting I hooked up the DVI connection as well and can easily switch back and forth from HDMI to DVI. The DVI connection by the way is perfect (infact, I would say beautiful...), with no distortion and no black borders. Anyway, I tried an older driver that came with the card as well as the latest one on ati.com, but still the same problem. I've been messing with settings for 4 days. I've tried every display mode and option in the ati software (catalyst). I haven't spoken with XFX yet because they seem to be not taking calls until January 4th...nice huh?...
BTW, I also hooked up my Playstation 3 to the monitor via HDMI, and the image seems perfect, no distortion or black borders. It definitely seems like an issue with the video card or software for the card. Anyway, I'll keep you posted. Please people, don't hold back if you hear anything...
On a side note, besides this one issue (which is hugely disappointing), I still think this is a pretty good video card. For the price < $200, I can't find one better with low power, 1GB gddr5, directx 11, up to 3 monitors, performs pretty well on the bench, life-time warranty with XFX, and looks awesome. That's why I haven't exchanged it yet.
Thanks.