The HD580 -- close cousin of the HD600 -- is just short of hard to drive, IMHO. You can't plug them into a truly weak source and get anything worthwhile out of them, but you also don't have to get a massively heroic amp to get decent sound out of them, earlier. There are other headphones commonly mentioned here that are truly difficult, but not the HD580/600s.
The more power you give them, the better, of course.
Let me give you an example: a CMoy amp is about as simple as solid state amps get. It's made of one IC, nine resistors and four capacitors. That's it. You can cram the circuit into a little over 1 square inch of board space without all that much trouble.
A Cmoy amp can power HD600s to plenty of volume, and the phones will have a good amount of body -- they certainly won't be thin. But, as you upgrade to faster and faster amps with better and better power handling, the headphones will gather clarity and snap in the mids and highs, and and tight, accurate bass.
This applies to all headphones, actually. It's just that headphones like the HD600 can do more with better headphone amps. Lesser headphones' performance almost plateaus at some point -- a better amp just won't help much with these other phones once you pass a certain point. There's little point in plugging in the freebie headphones that come with your portable CD player into a $1000 amplifier, but an HD600 will love the better amp.
These are all generalizations. Some amps mate better with HD580s/600s better than others. Again, just like all headphones and amps.