Yeah, the Roman Empire did have trade relations with nations towards the east, & that includes both land & naval trade. The silk routes for instance are well known for enabling trade & cultural transmission between ancient civilizations even before Rome. The Persians, Hellenes, Romans, nations from the Sub-Continent, et cetera traveled through these routes in order to establish political & economic relations. Furthermore, the Roman empire obtained lots of wealth alongside access to naval trade routes through the Red Sea after their conquest of Egypt. After the division between the Eastern & Western empire, one of the causes behind the Western empire's economic woes was that it had to somewhat fend for itself & was unable to benefit a lot from the wealth in the eastern region.
Apart from the presence of Roman coins in India, I think there may have been some Roman settlers present in a few regions too. The Mediterranean Sea was crucial to trade as well, & the desire to control those routes contributed to the start of the Punic wars. In any case, spices, perfumes, animals, & silk are just a few examples of items that were traded in the past. The purple dye that is generally believed to have been introduced by the Phoenicians was quite popular among the elites in different societies & wearing clothes dyed with that color was sort of a status symbol. By the way, diseases were another "commodity" of trade in the past. The Bubonic plague or the Black Death, which is essentially one of the greatest pandemics in human history, arrived to the Byzantine empire through the silk routes.
