It sort of depends. For one, the Mongol Empire only really existed as a unified entity until around 1260 when it became fragmented into the Ilkhanate(Persia), the Golden Horde(Russia/Central Asia), what would become the Yuan Dynasty(China, fully conquered by 1279). However, people in all of these areas would have known that they were under Mongolian rule for several reasons. The first, most obvious reason is that the Mongol would have destroyed their army, slaughtered the populations of many cities and laid waste to fairly significant portions of the countryside at which point they would be in charge and instituted new taxes. These taxes would have been new and in addition to all of the old taxes and were as exploitative as possible. The Mongols themselves were also cognizant of the fact that they had gained their empire through force and weren't appreciated by the peoples they had conquered, this led them to purposely try to remain separate and not assimilate in order to maintain their way of life and military dominance. Oftentimes Mongolian was made the official governmental language even though they allowed the government apparatus to continue as it was, often with indigenous bureaucrats in Persia and China.

As time went on,the extent to which the Mongols administered their subjugated territories varied from place to place. In Russia, the Golden Horde was still mainly situated on the steppe, and to a large degree maintained their old way of life, simply taking tribute from the nearby Russian princely states rather than directly administering them until they were eventually driven out by the Russians/destroyed and absorbed by Tamerlane. While they did not integrate with the Russians,they did assimilate with the Turkic peoples in that area of Central Asia. The Ilkhanate, which was more directly administering its Persian territory eventually assimilated to a much greater degree, while Mongolian was the official language, and Buddhism the quasi-official religion(Mongols allowed religious freedom but supported Buddhism) for quite some time, eventually the Mongolians converted to Islam and fully assimilated into the Turkic population that had already lived in Persia for quite some time (the Seljuk Turks had been in charge until about the time when the Mongols took over).When the Ilkhanate collapsed their was no mass exodus of the Mongol population,at that point they were fully integrated.In China,while the old bureaucracy was maintained and men like Khubilai were fairly sinified,the Mongols still maintained their separate language and culture and specifically used non natives in their highest government posts, the locals would have been entirely aware of them as a foreign ruling class. When the Ming took over,many Mongols returned to Mongolia, but many also remained and served as a cavalry force for the Ming Emperor,they were still a distinct ethnic/language group.

The Mongols never really settled the areas they conquered with large colonies like Greece or Rome, but they did eventually blend with the locals in some places. They never really built any monuments of a unique character or left a specifically Mongol cultural impact, their skill and importance lay more in their ability to take good ideas from different parts of their empire and use whatever worked and also allow trade between the different parts of their empire to flow uninhibited, bringing cultural influences across a vast area.

I hope this helps,sorry if it was rather long or confusing.

Source: Mostly David Morgan, The Mongols