From what I understand missionaries we're often the first wave of non-native speakers to learn Chinese or Japanese fluently. Usually it was to proselytize. I know that they also translated the Bible into Chinese as well, though I can't remember exactly one. What were the early struggles for these translators like? Did they just learned through trial and error? Was there a long periods of concentrated study in a university like setting? Or was it done over the course of their time spent in China interacting with the natives? Did they have any teachers?
Are you allowed, for instance, to swap "the Messiah", for the sake of clarity, to "the Mandate of Heaven"? The first major figure dabbling in questions of this sort has been the Italian 16 century missionary Matteo Ricci. He was received well at the imperial court and would both translate European classics (such as euclid's elements) into chinese and chinese classics (such as the analects) into latin. His attempts to dress christianity into the garb of chinese culture were popular with the chinese but they were also considered heretical by some of the other jesuits. The Vatican would often change sides on this issue, as they had to find the balance between staying orthodox and keeping the mission in china successful. When the religion got banned in 1724 there have already been 200,000 Catholics in China.
Now: the catholics did not translate the bible. For them the issue was about proselytising right.
During the century of humiliation the first protestant missionaries arrived with various different translations and their missionary efforts also contributed to one of the bloodiest conflicts in history: to the taiping rebellion. A certain Hong Xiuquan picked up christian scriptures from American baptist missionaries in Canton (allowed to operate there with impunity by the Treaty of Wanghia), came to the conclusion that he has also been a son of God (rather than just a dropout from the imperial examinations) and went on to proselytise in the mountainous Guangxi province. The government's attempt to suppress him went completely awry and ultimately resulted in a civil war that took tens of millions of lifes.
Source: A Star in the East. The Rise of Christianity in China by Rodney Stark and Xiuhua Wang.
Translation theory in general is not all that dependant on the exact pairs of languages. The issues that plagued Saint Jerome when translating to Latin where not much different from the issues addressed by Martin Luther more than a 1000 years later and the Chinese translators of the Buddhist sutras have been talking of all the same stuff. Modern translation theory prefers the summary of all these issues done in a famous essay by Friedrich Schleiermacher: either you bring a text closer to the readers (and their culture) or you bring the readers closer to the text (and the supposed culture of the text). In any case you'll have to make presumptions about it's meaning.
Source: Translation Theory by Jeremy Munday.