When searching for this it yields how to do this using ECS. This requires either putting a config file in a particular directory or setting env variables, neither of which are possible when using Fargate as it's serverless.
I feel as if I'm missing something here, anyone have any ideas?
Thanks!
update As of 2018 This answer is no longer valid; You can now use private registries. see https://stackoverflow.com/a/52267257
I found http://docs.aws.amazon.com/AmazonECS/latest/developerguide/task_definition_parameters.html#container_definitions
Which seems to imply it's currently not supported.
The easiest workaround would be to upload a copy of the image to ECR.