The Cloud SQL Auth proxy is the recommended way to connect to Cloud SQL, even when using private IP. This is because the Cloud SQL Auth proxy provides strong encryption and authentication using IAM, which can help keep your database secure.
When you connect using the Cloud SQL Auth proxy, the Cloud SQL Auth proxy is added to your pod using the sidecar container pattern. The Cloud SQL Auth proxy container is in the same pod as your application, which enables the application to connect to the Cloud SQL Auth proxy using localhost, increasing security and performance.
As sidecar is a container that runs on the same Pod as the application container, because it shares the same volume and network as the main container, it can “help” or enhance how the application operates. In Kubernetes, a pod is a group of one or more containers with shared storage and network. A sidecar is a utility container in a pod that’s loosely coupled to the main application container.
Sidecar Pros: Scales indefinitely as you increase the number of pods. Can be injected automatically. Already used by serviceMeshes.
Sidecar Cons: A bit difficult to adopt, as developers can't just deploy their app, but deploy a whole stack in a deployment. It consumes much more resources and it is harder to secure because every Pod must deploy the log aggregator to push the logs to the database or queue.
Refer to the documentation for more information.