Hello Garrette,
The Person Responsible field at project header level represents the project manager and the project manager can define the persons responsible at project task level.
Time recordings are sent to for approval to the person responsible of tasks, this selection is inherited to subordinate project tasks.
The task responsible person can take the decision to complete the respective assigned task if the work is completed.
Always the notifications will be sent to project manager but i am not sure will the person responsible will get any notification.
Best regards,
Sahalya