It really depends from whom you buy it from.
Good refurbishers will mention the original defect of the monitors or the category of defect (i.e. scratched screen, cracked casing, changed fuse, etc...).
As you say, it's a gamble and it's also a good idea to check out what kind of guarantee there is on the monitor. Is it 30 days or 3 months and such.
Personnaly, I own a refurbished HP monitor and the only defect is some sort of barely noticable scratch mark in the center as if it rubbed on something (more like a permanent smudge). I could probably fix it with some compound but it never bothered me.
The only thing that bothered me was the shipping. Ok, so it wasn't the shipping by itself but the duties to get the monitor from the US to Canada. UPS is f***ing expensive with respect to duties.