Did the people of Middle-earth know of the lands of old that sunk west of the Blue Mountains, or was it just not common knowledge?
For example: If by chance somebody brought up Beleriand, would people know of it?
Elrond remembered the battles of thangorodrim at the end of the first age that led to the destruction of Beleriand.
Thereupon Elrond paused a while and sighed. ‘I remember well the splendour of their banners,’ he said. ‘It recalled to me the glory of the Elder Days and the hosts of Beleriand, so many great princes and captains were assembled. And yet not so many, nor so fair, as when Thangorodrim was broken , and the Elves deemed that evil was ended for ever, and it was not so.’ ‘You remember?’ said Frodo, speaking his thought aloud in his astonishment. ‘But I thought,’ he stammered as Elrond turned towards him, ‘I thought that the fall of Gil-galad was a long age ago.’ ‘So it was indeed,’ answered Elrond gravely. ‘But my memory reaches back even to the Elder Days. Eärendil was my sire, who was born in Gondolin before its fall; and my mother was Elwing, daughter of Dior, son of Lúthien of Doriath. I have seen three ages in the West of the world, and many defeats, and many fruitless victories.
There are older elves than Elrond in middle earth (Cirdan, Galadriel and Celeborn for instance)
In the fictional history Bilbo is the author of the Silmarillion. This tells the tale of Beleriand so he must know.
He sings of avernien (the coastlines of Beleriand) before the Council of Elrond.
Eärendil was a mariner that tarried in Arvernien; he built a boat of timber felled in Nimbrethil to journey in; her sails he wove of silver fair, of silver were her lanterns made, her prow he fashioned like a swan, and light upon her banners laid.
It seems he also passed on this lore to Frodo, as he is aware of the history of the sword Sting
‘That would not help us now,’ said Frodo. ‘Come! Let us see what Sting can do. It is an elven-blade. There were webs of horror in the dark ravines of Beleriand where it was forged. But you must be the guard and hold back the eyes. Here, take the star-glass. Do not be afraid. Hold it up and watch!’
Aragorn is descended from Luthien and knows the lore, he will have learned it from Elrond, he certainly knows of Dorthonion which was immediately north of Berleriand.
The history of the Barrowdowns is discussed in the appendixes of the Lord of the Rings, it seems the Dùnedain kept the history of Beleriand alive.
‘It is said that the mounds of Tyrn Gorthad, as the Barrow-downs were called of old, are very ancient, and that many were built in the days of the old world of the First Age by the forefathers of the Edain, before they crossed the Blue Mountains into Beleriand, of which Lindon is all that now remains.
Tom Bombadil tells the hobbits of the world when it was younger.
When they caught his words again they found that he had now wandered into strange regions beyond their memory and beyond their waking thought, into times when the world was wider, and the seas flowed straight to the western Shore; and still on and back Tom went singing out into ancient starlight, when only the Elf-sires were awake.
Treebeard sings of Belariand
In the willow-meads of Tasarinan I walked in the Spring. Ah! the sight and the smell of the Spring in Nan-tasarion! And I said that was good. I wandered in Summer in the elm-woods of Ossiriand. Ah! the light and the music in the Summer by the Seven Rivers of Ossir! And I thought that was best. To the beeches of Neldoreth I came in the Autumn. Ah! the gold and the red and the sighing of leaves in the Autumn in Taur-na-neldor! It was more than my desire. To the pine-trees upon the highland of Dorthonion I climbed in the Winter. Ah! the wind and the whiteness and the black branches of Winter upon Orod-na-Thôn! My voice went up and sang in the sky. And now all those lands lie under the wave, And I walk in Ambaróna, in Tauremorna, in Aldalómë, In my own land, in the country of Fangorn, Where the roots are long, And the years lie thicker than the leaves In Tauremornalómë.
Thanks to an excellent comment, the various Maiar still around Middle-earth in the third age would have had knowledge of or even direct experience of Beleriand.
Sauron and the Balrog were around in the third age, Gandalf and Saruman and the other Istari would have had lore from various sources, even if they never personally visited it in their previous forms.
Old man willow and his kin where ancient, they seem to have memory of those days.
It was not called the Old Forest without reason, for it was indeed ancient, a survivor of vast forgotten woods; and in it there lived yet, ageing no quicker than the hills, the fathers of the fathers of trees, remembering times when they were lords. The countless years had filled them with pride and rooted wisdom, and with malice. But none were more dangerous than the Great Willow:
The dwarves seem to have poems and songs relating the first age so some clearly had knowledge of Beleriand. Indeed Gimli speaks of this in the Chamber of Mazarbul.
The world was fair, the mountains tall, In Elder Days before the fall Of mighty kings in Nargothrond And Gondolin , who now beyond The Western Seas have passed away: The world was fair in Durin’s Day.