I'm aware that the Chinese were aware of the existense of the roman empire during its height and that the Abbasid caliphs had to fight the Tang dynasty in the battle of Talas, but we don't generally hear much of the general relationship between the two sides of the world.
Did the average scholar or noble in Europe or the Middle East was aware of the Chinese empire's existense and vice-versa?
Yes they were both aware.
Have a read into Rabban Bar Sawma.
I couldn’t tell you enough about him myself but there is a lot of documentation of his travels between China and Europe during the medieval period which give a great insight into geo political divides, the spread of languages, and historical events and architecture.
He met with Kings, Priests, the Pope, Noblemen etc. The common language between Europeans and Chinese was Persian. Sawma was a seriously chill dude.