In particular, I'm interested in how it was seen by the Europeans and the Chinese. Did they remark on the unusual nature of this arrangement? Would official correspondence be addressed to the emperor or the shogun?
You don't really specify an era, so I'm just gonna talk about the 19th century and the Perry expedition as an example.
While planning for the expedition, the US understood Japan as having two emperors, a religious emperor and a military emperor. They addressed the letter it was the mission of the expedition to deliver to the "Japanese Emperor", with the intent of giving it to the Shogun in Edo. This lead to some confusion at first. They would become more familiar with the situation given some time after the opening of the country.