Stdout and stderr

I have a Pod with two containers.

apiVersion: v1
kind: Pod
metadata:
  name: test
spec:
  containers:
  - name: my-container
    image: google/my-container:v1
  - name: third-party
    image:  google/third-party:v1

One container is my image and the second is dynamic.
Inside “my-container” I want to collect all the stdout and stderr from the “third-party” container, add some metadata and write it with my logger.
I cant use a privileged container with volumeMounts.

If I could do something like this.

 containers:
  - name: my-container
    image: google/my-container:v1
    volumeMounts:
    - name: varlog
      mountPath: /var/log

  - name: third-party
    image:  google/third-party:v1 
    stdout: /var/log/stdout
    stderr: /var/log/stderr

 volumes:
  - name: varlog
    emptyDir: {}