Probably the most famous champion of the 2 amp setup is Billy Sheehan. There's plenty of Youtube vids / rig rundowns where he explains it in detail and why he does it.
He has 2 totally separate signal chains. One comes from the EB0 position pickup on his Yamaha Attitude and stays clean, keeping the low frequencies rumbling away, and the other signal comes from the split-P pickup on the bass and has little / no bass frequencies at all, and this is where he adds the distortion / other effects.
I did something similar for a while but by having a stereo pedal board, feeding 2 inputs on our mixer.
It works really well for not losing your low end when bass distortions are used.
I do wonder with all the current crop of expensive bass distortions around (Darkglass etc) whether it'll even be an issue in the future..