Does the government automatically receive documents sent over Peppol?

No. Documents only go to the addressed participant.

No. The Peppol network is not a central government hub. Documents are exchanged point-to-point between the sender and the verified recipient via their Access Points. The government does not receive a copy unless it is itself the addressed recipient (e.g. a government department).

In countries with VAT or reporting requirements (e.g. e-reporting/e-invoicing regimes), that reporting is done via the designated channels and rules. Peppol can be a transport channel there, but sharing with the government does not happen automatically.