I can give an answer from the Latter Han and three kingdoms of China perspective (25 to 284 AD) but I can't from the Roman perspective:

The Han knew Rome (or the Daqin as they called it) existed and in 97, the Protector General of the Western Regions Ban Chao sent staff officer Gan Ying, on Ban Chao's initiative, to seek out the Daqin. Gan Ying got as far as the Persian Gulf but there he was warned it was too long and dangerous a journey by shop by his hosts (who happily did the land journeys themselves but neglected to mention that route), instead he collected what he heard about the Daqin before his return home.

The only Roman envoy to the Hans happened under Emperor Huan in 166, arriving via ship at Rinan as had envoys had from India previously. However, there are doubts as to if it was an actual official envoy from Rome since Rome has no record and it is argued it might rather have been an enterprising merchant trying his luck. The King of Rome Andun (possibly Marcus Aurelius Antoninus) via his "envoy" paying homage to the Son of Heaven, bringing (notably disappointing) tribute and in exchange getting some gifts from the Son of Heaven who was showing his wealth and generosity. It suited Emperor Huan to boost his prestige amidst his quarrelling court and domestic difficulties.

China imported various types of coloured glass from Rome while during the civil war following the collapse of the Later Han, a merchant called Qin Lun (probably not his actual name) arrived at Jiaozhi and was sent to the court of Wu Emperor Sun Quan where he saw captives from rising general Zhuge Ke's campaigns in the south. This would also have been useful for Sun Quan's prestige as a founding Emperor.

In the Wei section of the SGZ, the primary source of the three kingdoms by Shu then Jin scholar Chen Shou, it ends with segments on foreign tribes and people. Thanks to the Wei scholar Yu Huan's Weilue, we have their understanding of the Daqin.

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Dàqín state is also called Líjiān, is at Ānxī’s, Tiáozhīxī great sea’s west, from Ānxī’s border Āngǔ city riding boat, straight through the sea west, if encountering favorable winds in two months arriving, if winds are slow maybe one year, if no wind maybe three years. Their state is at the sea’s west, therefore commonly they are called Hǎixī. There is a river going out of the state, west also there is a great sea. Hǎixī has [Wū]chísǎn city, from the state down directly north to Wūdān city, southwest again crossing a river, riding boat one day then crossing. Southwest again crossing a river, one day then passing. Altogether there are great capitals of three. However from Āngǔ city on land road directly north traveling to the sea’s north, again directly west going to the sea’s west, again directly south going to Wūchísǎn city, crossing one river, riding boat one day then passing. Circling back around the sea, altogether then crossing the great sea for six days then arriving at their state.

The state has small cities and towns altogether of over four hundred, east west south north it extends several thousand lǐ. Their King’s capital is on the shore beside the [Tiber] river and [Mediterranean] sea, with stone making the city wall. Their lands have pine, cedar, sophora, catalpa, bamboo, reed, poplar, willow, parasol tree, all kinds of plants. The people customarily farm and plant the five crops [millet, bean, sesame, barley, rice], raise and ride horses, mules, donkeys, camels, silkworms. Their customs have many unusual performances, mouth breathing fire, self binding and self releasing, juggling twelve balls with ingenuity.

Their state has no constant ruler, if in the state there is unusual disaster, then they change and establish a worthy man as the King, and release alive their former King, and the King also does not dare complain. Their common people when grown up are fair and upright, resembling the central state peoples but with Hú clothing. They say they were originally with the central states united but separated, and always wished to connect with envoys to the central states, but Ānxī plotted to take advantage, and they could not obtain passage. Their customs can in Hú write. Their government and public and private palaces and households have multiple floors, flying banners and beating drums, white canopy small chariots, and postal relay stations are installed like the central states.

Their state installs Lesser Kings of several tens, their King’s governing city’s circumference is over a hundred lǐ, and has government office departments of writing and documents. The King has five palaces, each palace from the other the distance is ten lǐ, their King at dawn goes to one palace to listen to affairs, reaching sunset and one night, and the next day he again goes to one palace, five days one cycle. They install thirty six officers, at each discussion of matters, if one officer does not arrive then they do not discuss. When the Kings goes out traveling, he always has accompanying men carry one soft leather sack and follow him, and those with reports, he receives their writings cast into the sack, returning to the palace then he examines and decides and reasons.

Using glass the make palace pillars and utensils. They make bows and arrows. Their separate branches are given fief of small states, called Zésǎn King, called Lǘfēn King, called Qiělán King, called Xiándū King, called Sìfù King, called Yúluó King,. The remaining small King states are very many, and cannot be one by one given in detail.

The state produces fine linen. It makes gold and silver coins, gold coins of one equal to silver coins of ten. They have weaving to make fine cloth, said to use river sheep hair, named as Hǎixī cloth. This state’s six domestic animals all come from water. Some say they do not only use sheep wool, and also use tree bark or wild cocoon silk, waving into woolen rugs, coarse mats, curtains and their sort all good. Their colors also are also brighter than what the sea’s east’s various states make. Also they always profit from obtaining the central states’ silk, unravel it to make Hú damask, and therefore repeatedly with Ānxī’s various states trade and market across the sea. The sea’s water is bitter and cannot be consumed, and therefore travelers hope to arrive in their states.

The mountains produce nine colors of less-than-jade stones, first is blue-green, second is scarlet, third is yellow, fourth is white, fifth is black, sixth is green, seventh is purple, eighth is red, ninth is violet. Now in Yīwú mountains there are nine colors of stones, that is of this sort. In Yángjiā Third Year’s time, [134] the Shūlè King Chén Pán presented Hǎixī’s blue-green stone and gold girdle one of each. Also the present Xīyù Jiùtú [‘Western Regions Old Map’] says Jìbīn and Tiáozhī various states produce

Dàqín has much gold, silver, copper, iron, lead, tin, divine tortoises, white horses, red manes, startling fowl rhinoceros, turtle shells, black bears, red immature dragons, avoiding poison rats, large cowries, tridacinae, cornelian, southern metal, agate, feather quills, elephant tusks, marked colored jade, bright moon pearl, luminescent pearls, true white pearls, tiger amber, coral; scarlet, white, black, green, yellow, blue-green, purple, misty, red, violet, ten kinds of circulating rarities; sparkling gems, white gems, water crystals, rose flowers, realgar, orpiment, green jade, five-colored jade; yellow, white, black, green, violet, red, purple-red, purple, golden yellow, misty preserved yellow, ten kinds of woolen rugs; five colored coarse carpet, five colored nine colored head lower coarse carpet, gold threat embroidery, various colored damask, gold scrawled cloth, purple chí cloth, fālù cloth, purple chíqú cloth, fire resistant cloth, āluódé cloth, bāzé cloth, dùdài cloth, Wēnsù cloth, five-colored peach cloth, purple-red earth gold woven canopy, five-colored dòu canopy, yīwēi wood, èrsūhé, dítí, mímí, dōunà, white aconite, xūnlù, yùjīn, glue rubber, fragrant vegetation of twelve kinds of perfumes.

Dàqín by road from the sea’s north on land can be connected, also by the sea and going south, with Jiāozhǐ’s seven prefectures the outer foreigners comparable, also there is water route connected to Yì Province and Yǒngchāng, and therefore Yǒngchāng produces rare things. Previous ages only discussed water routes, and did not know there were land routes, now they are summarized like this, their people and households are many and cannot be given in detail. From Cōnglǐng west, this state is most large, the installed various lesser Kings are very many, and therefore one records their dependents that are large.

In then goes into Daqin's relations with local dependents/powers.

Sources: Fire over Luoyang and Generals of the South by Rafe De Crespigny

SGZ 30.14 by Chen Shou with annotations by Pei Songzhi (in this case, Yu Huan), translated by Yang Zhengyuan