...they weren't?
That we don't know what was traded exactly (some people say it's the Cairo clock, but it is unclear and probably false) doesn't matter. The fact that Champollion already translated the rosetta stone from ancient greek and that hieroglyph could now be translated probably had an impact on this decision, especially since among all the Obelisks in egypt, Champollion singled out this particular one.
The empires pillaged it and gifted it among themselves.