The answer to your question is both "yes" and "no." The Stasi's operations were both public and secret. In fact, its power depended to a large extent on the tension and balance between publicity and secrecy: The Stasi needed to keep its operations confidential, but it also needed the public to know enough about them to be afraid. While the particulars of this or that Stasi operation - the identities of targets and informants, methods of surveillance, goals and protocols - were often closely guarded secrets, the general facts of the Stasi's existence and activities were not.

The Ministry for State Security (Ministerium für Staatssicherheit, commonly called the "MfS" or "Stasi") was a public organization. Its leader for most of the GDR's history, Erich Mielke, was one of the most recognizable men in the country and eventually a member of the Politburo of the Socialist Unity Party. Some Stasi leaders, by constrast, managed to maintain a degree of anonymity. Markus Wolf, the head of the Stasi's foreign intelligence operations, was so shadowy a figure until the late 1980s that westerners sometimes referred to him as "the man without a face."

At lower levels of the hierarchy, anonymity was also possible, but not complete or universal. In larger cities like Berlin, Dresden, Leipzig, or Magdeburg, Stasi employees could easily blend into the crowd. And the identities of the hundreds of thousands of informers who fed human intelligence to the Stasi were closely guarded secrets. Elsewhere, however, such as in the district offices of the Stasi scattered across East Germany, the identities of Stasi employees were often common knowledge. Historian Gary Bruce:

In smaller communities ... many townspeople would have known who was a full-time Stasi worker (though knowing the identity of an informant would have been considerably more difficult) and would have assiduously avoided coming into contact with them.

Nevertheless, according to Bruce, "many East Germans had difficulty forming an impression of the individuals who worked full time for the Stasi." This paradox - that East Germans often knew which people worked for the Stasi, but had no distinct "impression" of those people - is perhaps another way to describe the "tension" I mentioned earlier. Everyone knew about the Stasi, but only vaguely; their activities backgrounded daily life, and could have a profound impact on how people behave, even if people did not understand precisely what Stasi officers were doing.

So, how exactly did the Stasi operate? While the Stasi worked closely with other state authorities, including the Volkspolizei and criminal prosecutors, it was certainly capable of doing its work independently, and it preferred to do so privately. Bugs and taps were planted, informers were recruited and regularly interviewed, and searches carried out by Stasi employees. This passage from the 1982 West German book Die DDR-Staatssicherheit by Karl-Wilhelm Fricke describes the practicalities of arrest by the Stasi (based on the testimony of former prisoners; this is my translation out of the original German):

If the "operational observations" and "conspiratorial inquiries" [surveillance and interviews with informers] have lead to a decision to make an arrest, the arrest is, as a matter of principle, executed by several MfS employees - at least two or three. Normally, they try to avoid public view. Arrests in the early morning hours are preferred, be it at the workplace of the person being arrested or in his apartment. Place and time are specified exactly in the arrest order.

Stasi officers conducting arrests would require that the arrestee come with them zur Erkärung eines Sachverhalts ("to clear up an issue," the East German equivalent of an American cop asking you to "come downtown to talk"). To avoid public attention, arrestees were often transported in drab, windowless Barkas B1000 delivery vans specially modified for the purpose of prisoner transport.

The Stasi held and interrogated its prisoners in its own detention facilities, such as the famous remand prison in Berlin-Hohenschönhausen. These facilities were not, however, penitentiaries for long-term punishment - they were, in fact, specially designed to apply maximum short-term psychological pressure on the prisoners that would lead them to provide useful information during interrogation, using techniques like isolation and sleep deprivation. (This is in contrast to the actual Stasi prison in Bautzen, which was a legitimate penitentiary for political prisoners and spies, among other "special" convicts. The majority of convicts in the GDR were held in prisons administered by the Interior Ministry, not the Stasi.)

Altogether, the disguise, secrecy, and independence generally served to create what we might think of as a "social shadow" for the Stasi which was, at times, much larger than life. Uncertainty about who worked for the Stasi, who informed for them, whether they were listening to your phone calls, had bugged your apartment, or were talking to your coworker about the things you said at work, was a part of ordinary life for all East Germans, even those secure in the knowledge that they were not among the Stasi's targets. Secrecy meant ambiguity - if you were never sure where the Stasi was, that meant that it could be anywhere, or everywhere, and that you should behave accordingly.

Historian Jens Gieseke has referred to a "a mentality of subservience" that was a consequence of the Stasi's apparent omnipresence. The power of the security apparatus was implicit, a "broad but hidden presence, 'like an itchy undershirt.'" Historian Mary Fulbrook hits a similar note: "No one and nowhere could be deemed to be safe; friendships inevitably had a shadow of distance and doubt. ... Many East Germans lived with a sense of oppression and fear, although - perhaps even because - they did not know the extent of surveillance and interference in their lives." Fulbrook also notes that this phenomenon was not universal, but in so doing she illustrates how the power of the Stasi (and the state more generally) nevertheless became the background of ordinary peoples' obedient lives: "From an early age, it was instilled into East German citizens that they must conform, obey, the written and unwritten rules of behaviour, and never stand out in any way that might attract attention. ... Th climate of fear was the outer parameter of existence within the total Überwachungsstaat; it did not have to be a feature of everyday life."

SOURCES:

Bruce, Gary, The Firm: The Inside Story of the Stasi (2010)

Fricke, Karl-Wilhelm, Die DDR-Staatssicherheit (1982)

Fulbrook, Mary, Anatomy of a Dictatorship: Inside the GDR 1949-1989 (1995)

Gieseke, Jens, The History of the Stasi (2001)