They finally added Verizon Wireless support, so I got to try it out.
Basically, they give you instructions to set up conditional call forwarding to their toll-free number. When you get an incoming call from a private caller, it first rings your phone normally, with the number hidden. When you reject the call, it sends it to TrapCall who gets the ANI and then calls you again, this time showing the number in Caller ID.
It works quite well. Of course, it is still useless when the ANI information is missing or incorrect, such as with Skype.