Direct interaction between the Romans and Chinese was rather rare mainly because of the operations of the Parthian Empire in the Middle East, which was hostile to Rome. Only during times of relative peace were the two cultures able to send embassies, though trade often traveled between the two through intermediaries like the Kushans.

The Romans and Chinese were fairly well-aware of each other's existence and they respected each other to an interesting degree. For example, the Chinese called the Romans "Daqin" which literally means "Great China", showing how the Chinese saw Roman culture as a reflection of their own greatness. The average commoner in either state would likely not have much knowledge on such faraway places, but the merchants who traded for their goods and the nobility who purchased their luxuries would at least have some passing knowledge of their existence. Rome valued Chinese silks, and (rather amusingly) the Senate had to pass decency laws regulating the nobility's wardrobes as they felt that women's dresses fashioned of Chinese silk were too revealing (see Seneca's Declartions for more info). China, in turn, prized high-quality Roman glassware and Byssus, a rare fabric also called "sea-silk" fashioned from a certain mollusk's hairs, in addition to asbestos-clothing.

While some Roman accounts for the Chinese "Seres" date as far back as Augustus, the first "true" embassy (according to John Hill's Through the Jade Gate to Rome) between both nations was likely during the reign of Marcus Aurelius, when some Roman sailors arrived in Han China bearing gifts collected from India and the like. The Chinese encountered Roman embassies intermittently after that, dating to as late as the 1000s, long after Western Administration of the Roman Empire had collapsed.

It's a bit tougher to ascertain what the peoples of each state thought of the other, considering how limited their contact was. Contemporary Chinese Historian Fan Ye gives a drippingly favorable of account of the Romans in the Hou Hanshu's "Chapter on the Western Regions" and talks about their non-hereditary monarchy, ethics, honesty, and abundance of precious metals and jewels. He even says that the Romans "resemble the people of the Middle Kingdom, and that is why this place is called Great China". I personally can't recall any firsthand Roman accounts of China, so if anyone else can clarify that, that'd be great.

Interesting tidbit: despite trading with one another extensively, Roman and Chinese knowledge of the other's geography was limited. The Romans thought that China was centered near modern Indochina (at least, that's what Ptolemy's Geography states) and the Chinese believed that the Roman capital was Antioch, as that was the major trade city with which they had relations.