You absolutely should be taking precautions to make sure that what you're doing on AWS is secure. And in fact erring on the side of using providers who host in European countries, preferably European organisations.
(addendum - I spent a lot of the early part of this year working on AWS-based data processing systems for a large bank, they took massive precautions with the transport and storage of their data within the AWS system, including IPSec overlays, 14 day maximum node lifetimes and various other things. I realise that at some point you're trusting amazon, but there's a lot can be done to avoid having problems in the first place. "Not sending data to places you don't absolutely have to" seems pretty basic)
Google Analytics is a certainty, and reasonable alternatives are available.
I think it shows a terrible attitude from the department responsible to use GA. Not everything should be done by the lowest bidder, regardless of the costs. Maybe I'll write to my MP...
[1] https://www.cloud.service.gov.uk