Everyone who knew Melian likely knew. She wasn't there in relative secrecy, like the Istari were. I don't think this is ever spelled out explicitly, because it's just a matter of presumption - she wasn't there in secret, or undercover. All following quotes from The Silmarillion:
It was under the teaching of Melian that the Sindar became great:
Now as has been told the power of Elwë and Melian increased in Middle-earth, and all the Elves of Beleriand, from the mariners of Círdan to the wandering hunters of the Blue Mountains beyond the River Gelion, owned Elwë as their lord; Elu Thingol he was called, King Greymantle, in the tongue of his people. They are called the Sindar, the Grey-elves of starlit Beleriand; and although they were Moriquendi, under the lordship of Thingol and the teaching of Melian they became the fairest and the most wise and skilful of all the Elves of Middle-earth.
We're also told that she taught the Dwarves "much they were eager to learn." Given Melian's famed knowledge of the world, surely anyone who didn't know might have taken the hint!
The power of Melian stayed the Sleep of Yavanna:
Though Middle-earth lay for the most part in the Sleep of Yavanna, in Beleriand under the power of Melian there was life and joy,
When Thingol commanded that Menegroth be built, it was made with remembrances of Valinor in mind that Melian told of to the artisans:
But the Elves also had part in that labour, and Elves and Dwarves together, each with their own skill, there wrought out the visions of Melian, images of the wonder and beauty of Valinor beyond the Sea. The pillars of Menegroth were hewn in the likeness of the beeches of Oromë, stock, bough, and leaf, and they were lit with lanterns of gold. The nightingales sang there as in the gardens of Lórien; and there were fountains of silver, and basins of marble, and floors of many-coloured stones. Carven figures of beasts and birds there ran upon the walls, or climbed upon the pillars, or peered among the branches entwined with many flowers. And as the years passed Melian and her maidens filled the halls with woven hangings wherein could be read the deeds of the Valar, and many things that had befallen in Arda since its beginning, and shadows of things that were yet to be.
I think this is a pretty solid advertisement they, hey, I'm not from around here! if you didn't know.
The narrator, at least, suggests that she was often styled Melian the Maia when discussed the Girdle of Melian.
Therefore he withdrew all his people that his summons could reach within the fastness of Neldoreth and Region, and Melian put forth her power and fenced all that dominion round about with an unseen wall of shadow and bewilderment: the Girdle of Melian, that none thereafter could pass against her will or the will of King Thingol, unless one should come with a power greater than that of Melian the Maia.
for she knew that a parting beyond the end of the world had come between them, and no grief of loss has been heavier than the grief of Melian the Maia in that hour.
We are also told that yes, other people did know of the Girdle of Melian, and it seemed to be common knowledge.
Therefore the march-wardens said to Are-dhel: To the land of Celegorm for which you seek, Lady, you may by no means pass through the realm of King Thingol; you must ride beyond the Girdle of Melian, to the south or to the north.
Turin also knew of the Girdle, though you would expect him to:
For if the Girdle of Melian be broken, then last hope is ended. Nay, it is better indeed as things be; for a shadow I cast wheresoever I come. Let Melian keep them! And I will leave them in peace unshadowed for a while.’
Galadriel certainly knew who Melian was:
And at tunes Melian and Galadriel would speak together of Valinor and the bliss of old; but beyond the dark hour of the death of the Trees Galadriel would not go, but ever fell silent. And on a time Melian said: There is some woe that lies upon you and your kin. That I can see in you, but all else is hidden from me; for by no vision or thought can I perceive anything that passed or passes in the West: a shadow lies over all the land of Aman, and reaches far out over the sea. Why will you not tell me more?’
The Band of Barahir believed Melian could hallow waters:
But the waters of Tarn Aeluin were held in reverence, for they were clear and blue by day and by night were a mirror for the stars; and it was said that Melian herself had hallowed that water in days of old.