No they do not, that is the entire beauty of the pull-through mirror. For user code, as long as they keep referencing only to Dockerhub images, nothing needs to be changed (edit: except Gitlab CI configurations using docker:dind, which needs to be informed about the mirror).
The only downside is, as said, that it can't cache third party repos (quay.io comes to mind for people involved in k8s). For these, one has to mess with the resolv.conf and self-signed HTTPS certs for the Docker registry mirror.
There'd need to be a way within docker to alias to the new URL so that what normally would go to docker hub ends up pulling from the mirror.