From 1926 to 1933, Droste studied at the Ruprecht-Karls-Universität Heidelberg, the Ludwig-Maximilians-Universität München, and the Friedrich-Wilhelms-Universität (later, the Humboldt-Universität zu Berlin). He received his doctorate in 1933, from the University of Berlin. His thesis advisor was Lise Meitner, who was an adjunct professor (nichtbeamteter ausserordentlicher Professor) at the University of Berlin and directed doctoral research in her own section at the Kaiser-Wilhelm-Institut für Chemie (KWIC, Kaiser Wilhelm Institute for Chemistry, after WW II the Max Planck Institut für Chemie - Otto Hahn Institut), in Berlin-Dahlem.
From 1933 to 1942, von Droste was a scientific assistant (Mitarbeiter) at the KWIC, where Otto Hahn was the director and Lise Meitner headed a department. While at the KWIC, Droste contributed to the German nuclear energy project, also known as the Uranverein (Uranium Club).
In December 1938, the German chemists Otto Hahn and Fritz Strassmann sent a manuscript to Naturwissenschaften reporting they had detected the element barium after bombarding uranium with neutrons; simultaneously, they communicated these results to Lise Meitner, who had in July of that year fled to The Netherlands and then went to Sweden. Meitner, and her nephew Otto Robert Frisch, correctly interpreted these results as being nuclear fission. Frisch confirmed this experimentally on 13 January 1939. Droste and Siegfried Flügge, an assistant to Hahn, independently also predicted a large energy release from nuclear fission.
From 1942 to 1944, von Droste was at the Reichsuniversität Straßburg (Reich’s University of Strassburg), which had been founded in 1941, in German occupied France. He held his position there until 1944, when the Allied military forces liberated Strasbourg from German occupation. From 1944 to 1945, he was at Walther Bothe’s Institut für Physik (Institute for Physics) of the Kaiser-Wilhelm Institut für medizinische Forschung (KWImF, Kaiser Wilhelm Institute for Medical Research; today, the Max-Planck Institut für medizinische Forschung), in Heidelberg.
After World War II, the denazification process began. When Droste faced the proceedings, he turned to Werner Heisenberg, a prominent member of the Uranverein, for testamentary support – a document known as a Persilschein (whitewash certificate). Heisenberg was a particularly powerful writer of these documents, as he had never been a member of the NSDAP, he had publicly clashed with the NSDAP and the Schutzstaffel (SS), and he had been appointed by the British occupation authorities to the chair for theoretical physics and the directorship of the Max-Planck Institut für Physik then in Göttingen. Heisenberg wrote the document. In February 1947, Droste also requested support from Meitner; for complex reasons, she provided a document carefully vouching for his behavior without commenting on his character. Hahn too provided a Persilschein for von Droste, but Hahn also sent a critical letter to von Droste.
From 1946 to 1951, von Droste was in the Physics Department of the University of Heidelberg. From 1951 until his retirement in 1973, he was employed at the Physikalisch-Technische Bundesanstalt (PTB, Federal Physical and Technical Institute), in Braunschweig. He also held a position at the Technische Universität Braunschweig from 1967 to 1972 and was Regierungsdirektor (Government Director) from 1951 to 1973.
The following reports were published in Kernphysikalische Forschungsberichte (Research Reports in Nuclear Physics), an internal publication of the German Uranverein. The reports were classified Top Secret, they had very limited distribution, and the authors were not allowed to keep copies. The reports were confiscated under the Allied Operation Alsos and sent to the United States Atomic Energy Commission for evaluation. In 1971, the reports were declassified and returned to Germany. The reports are available at the Karlsruhe Nuclear Research Center and the American Institute of Physics.