My experience and opinion says they sound close to on par with the Shure 840 just voiced differently. The drivers are very similar and possibly the same. The construction of the inner cups is quite similar. Even the build quality. Since the 840 can be found for under $150 and the 940 under $200 I'd say that statement isn't far fetched. I can say they don't sound like or compete with $1k headphones. IMHO. Comparing them to headphones that have street prices between $500-$750 outclass the Shures. So you decide what that answer is.
I see. I was comparing headphones from their MSRP rather than their street price (as I usually do). That means I've listened to even fewer $150 USD headphones if you factor in street priced headphones.
I'm listening to my 940's through my iPod Touch (2G) at the moment and they work fine volume-wise (I'm in a quiet environment, volume level 1 with a volume limit at around 45%). I couldn't tell you if an amp helps since I don't own an amp at the moment. If you're wondering, I'm listening to some 320 kbps percussion and pop tracks at the moment.