There’s no concept of a “current” registry – full image tags always contain the registry address, but if no registry is specified then the Docker Hub is used as the default.
So docker push user/app pushes to Docker Hub. If you want to push it to a local registry you need to explicitly tag it with the registry address:
docker tag user/app localhost:5000/user/app
docker push localhost:5000/user/app
If your local registry is secured, you need to run docker login localhost:5000 but that does not change the default registry. If you push or pull images without a registry address in the tag, Docker will always use the Hub.
This issue explains the rationale.