As viewers we understand that it is the base of operations for The Faceless Men who serve the Many-Faced God. However they perform their duties secretly and their actions are unbeknownst to the outside world. However the House of Black and White seems to be a fairly prominent structure in the city, so how do the residents of Braavos feel about it? As in, do they understand who resides there and what goes on inside the walls?
Answers from both the show and books are acceptable!
Their individual identities are the secret, not their existence.
The Faceless Men were known assassins. Even considered by Westerosi as an option for hire.
Ned bowed, and turned on his heel without another word. He could feel Robert's eyes on his back. As he strode from the council chambers, the discussion resumed with scarcely a pause. "On Braavos there is a society called the Faceless Men," Grand Maester Pycelle offered.
"Do you have any idea how costly they are?" Littlefinger complained. "You could hire an army of common sellswords for half the price, and that's for a merchant. I don't dare think what they might ask for a princess."
A Game of Thrones - Eddard VIII
Residents of Braavos come to the House of Black and White to pay their respects and drink from the well. In Braavos death is not to be feared, it is welcomed. Consider that a greeting exchange is "Valar Morghulis" followed by a response of "Valar Dohaeris" (All men must die; All men must serve".
Worshipers came to the House of Black and White every day. Most came alone and sat alone; they lit candles at one altar or another, prayed beside the pool, and sometimes wept. A few drank from the black cup and went to sleep; more did not drink. There were no services, no songs, no paeans of praise to please the god. The temple was never full. From time to time, a worshiper would ask to see a priest, and the kindly man or the waif would take him down into the sanctum, but that did not happen often.
A Feast for Crows - Arya II
The city of Braavos was founded by ex-slaves, the first "faceless man" was also a slave that brought the "gift of mercy" to those who were suffering and eventually to those who caused the suffering.
The dead were never hard to find. They came to the House of Black and White, prayed for an hour or a day or a year, drank sweet dark water from the pool, and stretched out on a stone bed behind one god or another. They closed their eyes, and slept, and never woke. "The gift of the Many-Faced God takes myriad forms," the kindly man told her, "but here it is always gentle."
A Feast for Crows - Arya II
Overall, I think the establishment was well received by the citizens of Braavos.
