I'm not using Facebook, but I would presume that it should be working properly being one of the stock apps you received with your phone. Why it doesn't work, I wouldn't know. Might be that some of the newer updates screwed it up.
Some solutions come to mind:
1. Try a Facebook replacement app - I've seen people mention FriendCaster on numerous occasions, but there are others.
2. Try a hard reset - reflash the stock ROM.
3. Flash a newer ROM - something along the lines of 2.2 or 2.3.
My recommendation would be that you flash a newer ROM along with installing a replacement app.
Device: Sony Ericsson Xperia X8
ROM: FroyoPro v2.5
Kernel: Stock (11w41)
Clock: 245/729 MHz, Smartass