ANBU are chosen by the leader of the village, after having been watched throughout the academy for their potential and personality. In konoha they are not recruited until the age of 13, save for very few exceptions.
The hidden villages are a military state. They are something of an army base, with civilians working necessary roles for industry/commerce and entertainment. The civilians are not aware of what the shinobi do and while there might be whispers, particularly among the civilians with shinobi spouses, there is nothing concrete.
Most missions delegated to shinobi would be considered in varying ways as 'covert'. Obviously lower ranking missions given to genin are the type that most civilians can hire as service and do not count. Those would be D and C rank missions. Higher than that, it's usually a government / civilian town or whatever requiring services or military specific operations. Generally speaking, these are missions which are classified to a specific rank, though it's not unheard of for ninja to talk among themselves about details. Everyone within the shinobi system has information in some way because they need to know or else the system doesn't work as well.
However, there are also plenty that cannot be spoken of or are specific to certain divisions and declassified to others on a need to know basis.
Jounin are usually given B - A rank missions and some can be considered covert and therefore completely classified.
The ANBU are a separate division entirely working only sometimes with the police force. Their identities are only known by the hokage and the other ANBU within their own teams. That said it's not unheard of for ANBU to speak among each other and therefore know the identities of the entire organization.
But they are a BLACK OPS division. Black ops divisions are not beholden to the organization they belong. They are meant to be as secret as possible as in, should a Konoha ANBU attack another hidden village, it was not Konoha that did it if discovered. They go under incredible lengths to cover their tracks. That means that despite all shinobi being under the same organization ANBU are not at all meant to be known outside of ANBU. Their identities are classified, their job is classified, everything about them is classified and they essentially become no one until they resign or are released from duty.
The tattoo is never seen by civilians or other shinobi. It is hidden. It acts as a kill switch. If an ANBU member is caught or attempts to defect, they are killed. There are ways to get around this, but generally speaking no one tries it.
They are given cover stories for their whereabouts. Some may choose not to have a life outside of ANBU at all due to the high risk nature of their positions.
They usually have another 'job' within the village to mask their position. Saburo works as courier nin for his clan and so uses that as a common excuse. It is not unusual for him to disappear without notice, however.
Clans are aware of who in their families are ANBU due to how often the hokage requests their presence.
All equipment, including masks, are to be left at the ANBU headquarters because they are identifiers.
ANBU are not allowed to be outside of HQ in partial uniform, and if in uniform must always wear the mask. Occasional mistakes are made due to missions or the aftermaths of missions.
NO ONE KNOWS THEYRE ANBU UNLESS YOURE IN ANBU. Or were in ANBU.
There might be some who know a little more due to the fact that they were scouted and then rejected for admission. These shinobi are sworn to secrecy and often kept an eye on so as not to give even the slightest indication.
@maskage u got anything to add ash feel free.