As a matter of fact they are noisy because of information compression. The noise is just hopefully completely masked by the music itself. That's the whole idea of lossy audio formats.
I'm more alert now so I'm going to try this again... and hopefully add enough information that others will find it useful.
Mp3 compression is based off of removing parts of the music that is masked by other parts to be inaudible. The scheme does not mask noise, it is removing parts of the waveform that are not audible to the human ear. The result of overly compressed music is not noise... it's lack of detail.
Another way that mp3 compression works is by filtering out high frequency (treble) and low frequency (deep bass) because our ears are much less sensitive to those ranges as compared to mid range, and we are less likely to miss it. And again the result is lack of detail, lack of sparkle and oomph. It sounds less detailed and lively... but it doesn't sound noisy.
So what is a compression artifact? When trimming the waveform at too low a bitrate too much is removed and the sound is distorted. Not noisy but distorted. A concrete example that most are familiar with is cell phone conversation compressed at 32 k. Good enough to make out what is being said and who you are talking to, but the voice doesn't sound natural, it doesn't sound noisy just a bit distorted.
Alot of streaming music is at 64k and you sometimes hear this distortion, but mostly it just sounds lacking in detail and dull.
At 128k many people start to hear the sound as transparent-- sufficient detail and lacking in these kinds of transients that you can hear at 64k. Not as dull either but it could still sound better. You need better gear to need better bitrate (as opposed to earbuds on mobile devices).
Many blind tests put 160-192 range as reaching transparency even on quality headphones, amps, speakers.
Now if you want to develop a discerning ear, stop listening to 320k mp3 and pretending that it sounds bad (it sounds great and nobody has ears good enough to distinguish between 320k and flac). Start listening to really low bitrate mp3s and work your way up. I really think that when people talk about mp3s sounding noisy they are just picking up the noise floor on their crappy phone.