I indeed did the Euromasters a few years ago (04/05), but I'm probably not representative of the Euromasters population. I also had a look at the Durham page, but there's not much information.
As some people have pointed out, the degree at Bath is quite research oriented, but I don't know how that compares to Durham.
What I would advise you to do is try to contact some former Euromasters students, through the Euromasters office at Bath, or through Facebook, Linkedin, the yahoo groups etc. Also try to know about the student intake (about 60 at Bath) if you think that would affect your experience.
One advantage of the programme at Bath is that it's probably more international (but I found the entry standards to be quite low) and that you get to move between at least two universities in Europe (although you might find you're not really treated like a proper student at some sites).
All the argument that's been going on about which university had the best reputation is also not valid here, as we're talking about postgraduate studies (some lower ranked universities have really good postgraduate programmes) and these two universities are virtually unknown in the rest of the world - at least in secluded France where I hail from.
I also understand you are American: I remember there was some sort of partnership between the UNC at Chapel Hill and Bath.