es, your security and safety should be paramount. Never, ever compromise it for the sake of accepting an appointment. It is not worth the risk. You have to screen every potential client until you are satisfied.
The methods that ladies use vary greatly. Some will just rely on a hobbyists TER credentials (# of reviews, white list referrals). Some ladies will accept employment verification. Some ladies require 2 references. Some ladies will use verification sites. As Swim mentioned, PM a few ladies in your area and see if they will discuss their screening methods.
As far as the negative response you received from these two potential clients, that is their problem, not yours. You are perfectly within your right to ask for a reference or two. Any potential client that is offended by you asking for a reference is NOT someone you would want to see. They should not have been offended by your request. If they didn't have references, there are other ways for them to have been screened (verification sites, employment info, etc.). Now, I am not sure what "background information" you asked them to provide in lieu of references. Beyond verifying their place of employment, no one is going to provide you with personal information. So be careful what you ask them to provide.
Also, you were correct not to accept an appointment from a client based on his reviews written years ago. If you are going to screen based on TER credentials, the hobbyists should have recent reviews, several white list referrals, and some board presence (i.e. posting/commentary).
Again, only you can determine your comfort level and whether you have screened enough. Never let your guard down and accept an appointment if you are not 100% sure in the quality of the gentleman you are about to see.