With respect to the Roman period, we have a few clues. The History of the Later Han Dynasty (Hou Hanshu), compiled in the fifth century, contains a brief passage on trade goods from Da Qin (The Roman Empire):
"This country produces plenty of gold, silver, and precious jewels, luminous jade, bright moon pearls, fighting cocks, rhinoceroses, coral, yellow amber, opaque glass, whitish chalcedony, red cinnabar, green gemstones, drawn gold-threaded and multi-coloured embroideries, woven gold-threaded net, delicate polychrome silks painted with gold, and asbestos cloth." (12)
An excellent commentary on this list can by accessed by clicking on the notes linked to this text.
Most of the products listed by the Hou Hanshu were (unsurprisingly) produced in the Roman east, and traded primarily by the merchants of Alexandria and Antioch. Some of them, like the rhinoceros horn and amber mentioned in the list, were not products of the Empire itself, but were handled by Roman merchants. Others were Roman products: that asbestos, for example, probably came from the mountains of Cyprus.
Another passage in the Hou Hanshu remarks:
"In the reign of Emperor Huan, king An-tun of Da Qin (Rome) sent an embassy....this offered ivory, rhinoceros horn, and tortoise shell....but their tribute contained no jewels" (88)
This was not a real embassy - as far as we know, in fact, the Romans never sent a formal delegation to China. It was almost certainly a party of Roman merchants operating between Alexandria and India, who had either been shipwrecked on, or attempted to trade along, the southern Chinese coast. The ivory and tortoise shell they brought were probably from India, though the rhinoceros horn may have been imported from sub-Saharan Africa via Alexandria.
In general, however, the Chinese do not seem to have been especially interested in most Roman products; Pliny the Elder remarks in his (admittedly very ill-informed) description of "the Seres" (Chinese), "they shun all intercourse with the rest of mankind, and await the approach of those who wish to traffic with them" (6.20).
Perhaps this answer by u/Mighty_Dighty22 to a similar question can shed some light while we wait for other answers.
According to Dmitry Voyakin, goods that are known to have traveled East along the Silk Roads include "frankincense and myrrh, jasmine and amber, cardamom and nutmeg, ginseng and bile of a python, carpets and fabrics, dyes and minerals, diamonds, jade, amber, corals, ivory... gold and silver bullions, fur and coins, bows, arrows, swords and spears" as well as animals and more abstract elements of culture such as fashion.
Chinese pottery was among the Eastern goods that traveled West along the Silk Roads. It may be of interest that some of that pottery may have been produced in the East using a material imported from the West. There is not currently a scholarly consensus around this possible Western export, cobalt blue pigment from Persia. This cobalt would have been used in pottery centres like Jingdezhen in Jiangxi province, which produced iconic blue-and-white pottery during the Ming dynasty (CE 1368-1644).
Support for the idea that the distinctive blue under-glaze was painted using imported Persian cobalt comes from studies which examine the manganese content in blue-and-white Ming artifacts. Chinese cobalt is normally thought to have high amounts of manganese, whereas cobalt ores found in Persia typically don't have manganese. Based on several different studies, some scholars divided the artifacts into two distinct groups. Samples from before CE 1425 have cobalt with very low manganese levels (and high iron), suggesting the pigment was imported from abroad; samples after CE 1425 have cobalt with significant quantities of manganese (and low iron), meaning their pigment was probably sourced from within China.
Those who are skeptical of significant amounts of blue pigment being imported to China from the West generally don't dispute the findings pointing to low manganese levels in the cobalt of some blue-and-white Ming ware. The point of dissension is the conclusion that these samples of cobalt cannot have come from inside China. One author, Adam T. Kessler, supports this view, in part, by suggesting that primary sources from the late Ming dynasty reveal that some cobalt historically mined in China may chemically resemble what is thought to be 'imported' cobalt. Kessler refers to an English translation of the Tiangong Kaiwu (originally written in CE 1637 by Song Yingxing), which mentions pigment being sourced from a city in Jiangxi named for its arsenic-richness, as one example of historical evidence contradicting the dominant modern understanding of which materials could have been used in production of wares.
While it is difficult to say with absolute certainty that before CE 1425, blue-and-white porcelain produced in Ming dynasty China owed its iconic blue under-glaze to cobalt that traveled along Silk Roads from Persia, it is widely believed to be the likely case. In addition to trade signifying exchange, it may have meant collaboration in this instance.
**^(Sources)**^(:)
^(Dillon, Michael. "Transport and Marketing in the Development of the Jingdezhen Porcelain Industry during the Ming and Qing Dynasties.") ^(Journal of the Economic and Social History of the Orient) ^(35, no. 3 (1992: 278-90.)) ^(JSTOR,) ^(www.jstor.org/stable/3632734)
^(Du, Feng & Su BaoRu. Science in China Series E-Technol. Sci. (2008 51: 249.)) ^(https://doi.org/10.1007/s11431-008-0013-0)
^(Juan, Wu, Pau L. Leung, and Li Jiazhi. "A Study of the Composition of Chinese Blue and White Porcelain.") ^(Studies in Conservation) ^(52, no. 3 (2007: 188-98.)) ^(http://www.jstor.org/stable/20619501)
^(Kessler, Adam T. "Song Blue and White Porcelain on the Silk Road" (2012)) ^(Studies in Asian Art and Archaeology) ^(BRILL: Leiden, Boston. P. 511.)
^("The Exploitation of the Works of Nature (Tiangong Kaiwu)," ^(World Digital Library)) ^(https://www.wdl.org/en/item/3021/)
^(Voyakin, Dmitry, "The Great Silk Roads." UNESCO.) ^(https://en.unesco.org/silkroad/content/great-silk-roads)
^(Edit: formatting struggles.)