I can give an answer from the Han perspective: Ban Chao's use of military force and diplomacy in the "Western Regions"/parts of Central Asia did bring a route for some diplomacy with far powers across the Silk Road for a brief time. An embassy and gifts from Kushan in 87AD with antelope (thought at the time to be a unicorn), 88 similar gifts from Parthians, in 90 Kushan's asked for a princess and military aid then attacked when refused but Ban Chao was able to see them off with a scorched earth strategy. In 101 Parthians brought lions and ostrich feathers which went rather better.

It isn't always clear if (bar the 90 one) if such embassies were from the rulers or whether they were from wealthy merchants and local groups seeking trade.

The Han knew Rome or the Daqin existed and in 97, the now Protector General 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 Han's control of the Western Regions and ability to get such embassies wouldn't last, the Han hadn't been too encouraging of Ban Chao's exploits at the best of times, often relying on allied troops and his own resources, while problems closer to home would occupy attention and resources of the court. By 106 the Dowager Deng ordered Protector General Duan Xi to abandon the Western Regions as a local revolt spread beyond their control while in 107 the Qiang revolt, which would last for ten years, meant the Han were cut off from the Western Regions. They would never regain full access and control with Ban Ying in the 120s only able to restore some limited return. Once the retreat of 106 began, I'm unaware of any contact with the Kushan empire or from Parthia.

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 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.

The Han were aware of the existence of other powers but the window for contact from the north was brief and Rome only contacted once (if it all) while the Han only attempted such an envoy once themselves due to the initiative of the commander on the ground. Diplomacy such as it was in those cases was usually a form of gifts and prestige from ruler to ruler or merchants to ruler.

Source: Fire over Luoyang by Rafe De Crespigny