I have done it *just once*, with the first agency I was trying to get verified by, and I did it through a bizarrely complicated, protracted and extremely indirect way, for which I am forever grateful to this particular agency for tolerating. I won't do it again, and my greatest hope is that this information was of only transient value to the verifier and that it was promptly deleted upon verification (of course, I have no way of knowing this, and so must I presume it persists on some unencrypted drive volume on some vulnerable PC or PDA out there somewhere (*shudder*).
As a newbie myself, I can say that this was without a doubt easily the most distressing part of this new experience, forfeiting my real identity information to completely unknown stewardship.
The girls will say we shouldn't be bothered by this because they have so much at stake. While it's true that the safety of these ladies is terribly important, for some of us gents, a bust would result in a huge, life-changing experience not just for ourselves, but possibly for many others as well.
Since that initial verification, I have been slowly building my scope of verified resources, both agency and independent, through TER reviews, references from providers, white-listing, DATE-CHECK references, etc. I think there is a point at which a gent begins to look like a fairly safe bet from the agency/provider perspective and the verification challenge diminishes somewhat. That said, there will still be agencies and providers who insist on the full ID disclosure and I will pass those up in favor of others who will rely on my alternate identity and on the track record I'm building.
I'll admit that maybe I'm a bit paranoid, but perhaps I have a lot to be paranoid about. For me, this advice to supply your real identity information *just once* was some of the very best advice I've received from the learned sages on this TER board. The fewer instances of your real identity out there with agencies and independents, the lower the probability of something bad happening with your personal identity information.
Having said all that, most verifiers simply want to determine if you are who you say you are and will go about contacting employers in a very discrete way. Remember, it is in their best interest also to be very discrete. Sometimes a confirmation from the receptionist that you're not in at the moment or are on another line, or a legitimate voice mail recording from your personal business phone extension will do the trick.
Also, that thing in your post about "well-reviewed providers", that also mitigates your risk of providing personal identity information enormously; there's a good reason why they are well-reviewed! Good luck!