I've personally called Shady and TVT...both pretty big labels who told me to send them shit. I've spoken to reps from Def Jam, Interscope, and J Records. Granted, you need a manager and/or attorney if the label sets up a meeting with you...but it's pretty simple to get the OK to send demos, you just have to be on top of your business game and speak to the right rep.
Maybe I've gotten through because I have my game right. I have a manager now, but when I was calling these labels, it was on the solo, and my efforts weren't fruitless.
UNSOLICITED means sending shit without notifying them first. That's all it is, nothing too complicated. You don't need a lawyer to solicit your shit. Can't really tell me I don't know what I'm talkin' about, 'cause I've done it. I also have my copyrights, own my masters and my own publishing through ASCAP...all of which I obtained without a manager or a lawyer.