Not sure about how much vetting goes on in these agencies. I think very little or none, especially with the agents who list Eastern European (EE) girls. But the agencies I've used frequently and found reliable in the past have been the following (I think girls through these agencies are vetted properly):
External Link/Members OnlyExternal Link/Members OnlyExternal Link/Members OnlyExternal Link/Members OnlyMost of the escorts on these sites are British and they are expensive. If you have time to search Adultwork you will also find many of the escorts listed on there for more reasonable price as independents. Saying that though, I have not had any disastrous punts from the girls listed through these agents, at worst average. Generally, I don't use agencies much these days.
I have also used the following escort agencies way back in the past who generally list Eastern European escorts (not used these in many years now). But, they do seem to be popular amongst other members on here:
External Link/Members OnlyExternal Link/Members OnlyExternal Link/Members OnlyExternal Link/Members OnlyThese are probably the most popular agencies in London for hosting EE girls, but there are many more out there. If you look at all the EE escort agencies you will realise that all of them stock the same set of girls. As I mentioned above, I think there is very little or no vetting with the EE girls listed through these agencies. They aren't cheap either as they once used to be, especially if you are after enjoyments like CIM which now frequently incurs an extra charge. You would have to be careful in selecting your girls through here since many of the girls tend to be unreliable / hit and miss Romanians (but all the agents dare not write the girl's true nationality on their profile, and will list the girl's nationality as anything but Romanian).