Starting 11th October 2021 COVID tests will no longer be free for residents who are unwilling to get vaccinated. However, for international students who have been vaccinated with a serum which is not approved in the EU the COVID tests will stay free until 31.12.2021, following §4a Coronavirus-Testing-Regulations of 21.09.2021.
Additionally, the Standing Committee on Vaccination recommends those students to get a second series of vaccinations with an EU-approved serum, but reminds of the possiblity of stronger reactions to the injection (Epidemiologisches Bulletin 38, 2021, Page 8). This second series is necessary to received the "vaccinated" status in the EU, otherwise individuals without EU-approved vaccination are regarded not vaccinated. You should mention your previous vaccination to your physician!