This source mentions a diplomatic mission to China arriving in 1081, and it made me wonder how they would have communicated. I suppose I'm also interested in the same question about other pre-modern societies that had little contact with each other yet still communicated in some way.
They never really met directly - i.e. Byzantine and Chinese diplomats never actually travelled to each other's territories. But they did meet through commercial contacts/trade routes in Central Asia. They simply knew of each other and each other's trade products.
For the Chinese, they were aware that if they kept going west, there was eventually one or more seas, but they may not have realized the Caspian, Black and Mediterranean Seas were all different. They knew there was a giant capital city out there, but they didn't really know exactly where Rome was, and probably not Constantinople either. The big city they were thinking of might have been something else, like Antioch or Damascus.
Same for the Romans/Byzantines. After Persia they really had no direct personal experience of anything further east, they just knew that they got valuable stuff from way out there somewhere. They had a vague idea that there were some deserts and some big mountains, and eventually a lot of islands, but they didn't know where the cities were or anything like that.
As for how they translated, the Romans/Byzantines didn't know Chinese and the Chinese didn't know Latin/Greek. They got all of their information from the various people who lived in between, who spoke Turkic languages, Persian, Aramaic, or Armenian. Central Asia was a huge mosaic of different cultures and there had been translators and interpreters there for hundreds, maybe even thousands of years. When Europeans and Chinese came into more direct contact in later centuries, they used translators and interpreters too.
We've had a few other recent questions about similar topics, so check out those for some more information, sources, and even more links to other questions:
Was there contact between the Byzantines and the Chinese?
How powerful were translators during ancient, medieval or early modern times?