It's complicated, so here's an over-simplification:
German archaeologist Ludwig Borchardt found the bust in Egypt in 1912, at a time when Egypt was a British protectorate. However, since Napoleon Bonaparte's occupation of France in 1798, the French had retained almost exclusive control of all antiquities in Egypt. The first clause of the 1904 agreement between France and England which partitioned North African into spheres of influence stated: "It is agreed that the post of Director General of Antiquities in Egypt shall continue, as in the past, to be entrusted to a French savant."
So, in 1912 (and in 1906, when George Herbert, Fifth Earl of Carnarvon, of King Tutankhamun fame, came to the Valley of the Tombs of the Kings), a private citizen only had to be alert and wealthy enough to find an "in" with an appropriate Frenchman in the Ministry of Public Works' Service of Antiquities to obtain a license or concession to dig - usually above the Theban Hills, west of the city of Luxor, in the vicinity of the Valley of the Tombs of the Kings. All work was carried out at the expense, peril, and risk of the concessionaire.
Under the terms of the concession, all mummies, coffins, and sarcophagi were to remain property of the State, as well as objects in any 'intact' tombs (no 'intact' tombs have been found - even Tuttankhamun's tomb had been been entered twice, although the thieves had stolen or damaged only a small number of items, and one group was apparently caught in the process).
Items found in any tombs "searched in ancient times" would be divided between the holder of the concession and the Antiquities Service. The Antiquities Service would take all works of capital importance from the point of archaeology and history and would share the remainder evenly with the permittee. This was all done in a fairly offhand manner up until the time of Tutankhamun's tomb. The Antiquities Service had nominal expertise of its own and often relied upon the concessionaire's expertise to decide what works were of capital importance.
So, back to German archaeologist Ludwig Borchardt. In his records, he described the bust of Queen Nefertiti as a 'painted wooded bust of a princess.' Given the way the archaeology game was played in the early twentieth century, Borchardt may well have known it was Nefertiti and called it a princess to downplay its importance. A Frenchman in the Antiquities Service determined it was not of capital importance in terms of history or archaeology. Borchardt formally received the bust in the division of spoils, signed by a Frenchman. He received an export permit, signed by a Frenchman, a Britisher, an Egyptian, I don't know.
In the early 1930s, Egypt demanded that the bust be returned. The early 1930s saw the rise of the Wafdist nationalists in Egypt, who demanded independence from England and the west. Germany agreed to return the bust. However, before the bust was returned, Adolf Hitler prevented its return. Some time before 2007, Egypt started to request a 'loan' of key objects, such as the Nefertiti bust, the Rosetta stone, a zodiac ceiling painting from the Dendera Temple now at the Lourvre, busts of Anchhaf and Hemiunu (a pyramid builder and the nephew of a pyramid builder). When the Berlin museum said the Nefertit bust was too delicate to travel, the head of Egypt's Antiquities Service said Egypt would seek legal recourse to prove the bust was stolen and to seek permanent return of it. Berlin believes that the request for a loan is just a ruse and that Egypt intends to keep all of the items it has requested.
So, in Egypt's behalf: Egyptians didn't decide to give the Nefertit bust to Borchardt, Frenchmen did; Borchardt may have misidentified the bust so that it wouldn't be claimed by a French bureaucrat in Egypt's Antiquities Service as a work of "capital importance; and Germany once agreed to return the bust, then reneged.
In Germany's behalf: Borchardt had a written concession from Egypt's official Antiquities Service; Borchardt paid for the excavation in which the Nefertiti bust was discovered; the bust was examined by an official from Egypt's Antiquities Service and officially awarded to Borchardt as part of the division of spoils; and Borchardt received an Egyptian export license for the bust.
Oh. Just to show that times haven't changed. The big kerfuffle that caused Howard Carter to close Tutankhkamum's tomb and walk off the site? Well, after he lifted the granite lid of Tut's sarcophagus to expose the first golden coffin, he asked for permission to allow the excavator's wives to enter the tomb to see the golden effigy. After all, some of them had been in Egypt for a decade, give or take, with their husbands. However, the new Under-Secretary of Public Works, Mohammed Pasha Zaghlul decided that it would not be acceptable for women (women!) to see the golden coffin before government officials. The Wafdist Under-Secretary was Muslim.
Thanks. I didn’t know all that. While I agree, national treasures should stay in their respective countries, since everyone signed off on the bust then it was Borchardt’s to do with what he wished. If the french savant or whomever wasn’t smart enough to do his job and properly identify the object, then that’s not Borchardt’s fault. Ah, well.