Docker images in private repos should have be accessed by the private repo's FQDN, for example:
When pulling by this image name, it's implied that the registry is at
docker.example.com and is accessible to the Docker host. It looks like the Docker module makes mention of API usernames and passwords, implying that it has the capability to authenticated against
hub.docker.io and any registry that uses the index/hub authentication. I'd have to look through the source code to be sure, but I think that's what it's doing. The docker image module doens't have this capability.
At any rate, check out what auth mechanism you're using on your private index, check that the docker host has access to the registry.
For what it's worth, I use the docker module to work against my private registry, and it works without issue.