http://depts.washington.edu/silkroad/texts/texts.html

I present to you a collection of sources collected regarding east-west knowledge and transit in that period. Here we have sources mentioning embassies visiting Parthia in 91 BCE and later making it all the way to Alexandria, Antioch and showing knowledge of the 'middle sea'

'When the first embassy was sent from Zhongguo [China] to Ar-hsi [Arsacids, or Parthia], the king of Ar-hsi ordered twenty thousand cavalry to meet them on the eastern frontier. The eastern frontier was several thousand li distant from the king's capital. Proceeding to the north one came across several tens of cities, with very many inhabitants, allied to that country. After the Han [Chinese] embassy had returned they [the Parthians] sent forth an embassy to follow the Han embassy to come and see the extent and greatness of the Han Empire. They offered to the Han court large birds'-eggs, and jugglers from Li-kan [Syria].'

(http://depts.washington.edu/silkroad/texts/romchin1.html)

And from a later account;

From the city of Ar-ku [Uruku, modern Warka] , on the boundary of Ar-hsi one takes passage in a ship and, traversing the west of the sea, with favorable winds arrives [at Aelana, modern Elat, on the Gulf of Aqaba] in two months; with slow winds, the passage may last a year, and with no wind at all, perhaps three years. This country is on the west of the sea whence it is commonly called Hai-hsi [Egypt]. There is a river [the Nile] coming out from the west of this country, and there is another great sea [the Mediterranean]. In the west of the sea there is the city of Ali-san [Alexandria]. Before one arrives in the country one goes straight north from the city of U-tan [Aden]. In the south-west one further travels by a river which on board ship one crosses in one day [again the Nile]; and again south-west one travels by a river which is crossed in one day [still the Nile]. There are three great divisions of the country [Delta, Heptanomis, Thebaid]. From the city of Ar-ku one goes by land due north to the north of the sea; and again one goes due west to the west of the sea; and again you go due south to arrive there. At the city of Ali-san, you travel by river on board ship one day, then make a round at sea, and after six days' passage on the great sea [the Mediterranean], arrive in this country. There are in the country in all over four hundred smaller cities; its size is several thousand li in all directions of the compass. The residence of their king lies on the banks of a river estuary. [Antioch-on-the-Orontes].

(http://depts.washington.edu/silkroad/texts/romchin1.html)

From these we get a picture of general mystery and unfamiliarity. It seems unlikely from this evidence that anywhere close to a majority of the people would know of the existence of places like China, let alone an accurate appearance of their people.

From the other side, the Periplus Maris Erythraei (or ‘Voyage around the Erythraean Sea’) gives a Greek speaking Egyptian's view of the route west-east instead at roughly the 1st Century C.E;

'64. After this region under the very north, the sea outside ending in a land called This, there is a very great inland city called Thinae [i.e. China], from which raw silk and silk yarn and silk cloth are brought on foot through Bactria to Barygaza, and are also exported to Damirica [=Limyrike] by way of the river Ganges. But the land of This is not easy of access; few men come from there, and seldom. The country lies under the Lesser Bear [Ursa Minor], and is said to border on the farthest parts of Pontus and the Caspian Sea, next to which lies Lake Maeotis; all of which empty into the ocean.

(http://depts.washington.edu/silkroad/texts/periplus/periplus.html) Emphasis mine.

This makes it extremely unlikely that trade and contact was frequent enough to give either group a good knowledge of each other.