When an escort asks you for personal information so she can verify that you're not a dangerous loon or a member of the law enforcement community, she's doing what is called "screening". Each lady has her own level of comfort with various screening procedures, but references from other escorts who have seen you are almost always the very best things to have. The one warning I'd put in here is that you should let a lady know if you plan to use her as a reference - she may need a quick refresher on just who you were, especially if it's been a few months since she saw you. I've found that saying, something like, "I'm so-and-so, and we met at this place on this day, and I recommended Restaurant X to you and told you the joke about American beer being like sex in a canoe (they're both fucking close to water)." It's also possible that the lady may have gotten out of the business for whatever reason, or simply be unreachable in the time frame that the new lady has for screening you.
I've heard of many ladies who use screening services like Date-Check or Preferred411, too. Some of these services will sell you a membership, whereupon they take care of screening you, so you just tell the lady, "My P411 handle is such-and-such", she calls them, they verify, and you're good to go.
I've never had a problem with giving any info to a lady, but I have seen news stories about escorts getting busted who have kept some records; often, the nature of the records is unclear in the news stories. It might be as simple as "Tim, email address is [email protected], 2-hour date on 4/1/05, no problems", or it might be as complex as "Billy Johnson, local politician, mole on ass, liked having his balls tickled, home phone is XXX-XXX-XXXX, Visa card is XXXX with expiration date yy/ZZ."
Just like ladies each have to make their own decisions about how much information they need, you have to decide how much information you're comfortable with giving out. It's a risk, but only you can judge exactly how big a risk it actually is. For me, I'd be willing to show my ID, but not a credit card; I've given work info, but I wouldn't give out a home phone. I've freely given out my TER handle, and I've got a Date-Check trial membership; it's doubtful that I'd ever prepay a deposit, though.
I totally agree with Misty Bay's point about not lying. If you don't feel comfortable giving her the information she asks for, just say so. It will probably end the chances of you meeting the lady in question, but that's better than the result if you go down that road and she finds out you have misled her.