Question: What is the easiest way to duplicate an entire Amazon S3 bucket to a bucket in a different account? Ideally, we'd like to duplicate the bucket nightly to a different account in Amazon's European data center for backup purposes.

Answer: 
One thing to consider is that you might want to have whatever is doing this running in an Amazon EC2 VM. If you have your backup running outside of Amazon's cloud then you pay for the data transfer both ways. If you run in an EC2 VM, you pay no bandwidth fees (although I'm not sure if this is true when going between the North American and European stores) - only for the wall time that the EC2 instance is running (and whatever it costs to store the EC2 VM, which should be minimal I think).
You can make an application or service that responsible to create two instances of `AmazonS3Client` one for the source and the other for the destination, then the source `AmazonS3Client` start looping in the source bucket and streaming objects in, and the destination `AmazonS3Client` streaming them out to the destination bucket.