My current approach is to have logging to stdout for each container and then get docker to send that to syslog, or to syslog to a remote syslog if the container cannot do stdout, and setup logstash to be that syslog sink.
Does AWS offer any facility for logging that I should be considering instead?
ECS integrates with Cloud watch logs.
You can configure your container instances to send log information to CloudWatch Logs. This enables you to view different logs from your container instances in one convenient location. This topic helps you get started using CloudWatch Logs on your container instances that were launched with the Amazon ECS-optimized AMI.
http://docs.aws.amazon.com/AmazonECS/latest/developerguide/using_awslogs.html