According to the German Missions in the United States, service of documents in cross-border legal relations is governed by the Hague Convention on the Service abroad of Judicial and Extrajudicial Documents in Civil and Commercial Matters.
In each country, a central authority has been charged with the formal service of documents. In Germany, the decentralized judicial authorities of the German federal states are the competent authority to effect service.
Service of U.S. or Canadian Documents in Germany
Official Service via German Authorities
The documents to be served shall be forwarded to the central authority in Germany on the respective federal state by the authority or judicial officer competent under U.S. law (this may be the attorney of one of the parties). The request for service must be accompanied by a summary of the document(s) which must be written in, or translated to, the German language. The summary and the document (including, if applicable, its German translation) will be served upon the addressee. This will be initiated by the competent central authority, or, upon its request, by the local court in Germany. The competent authority will issue a certificate of service and forward this to the authority or person having requested the service in the other state.
Direct service of U.S. document in Germany via postal channels is not permitted, as Germany has raised objections under Article 10 of the Hague Convention.
Source: German Missions in the United States
"Service of Documents Between the US and Germany"
https://www.germany.info/us-en/service/08-Documents,CertificationsandApostille/service-of-documents/945230