It has become increasingly unclear if the Titans are meant to be a publicly operating superhero team or if they are a covert team that is incredibly bad at hiding themselves from the public. This point is more confused than ever following the latest episode, "Faux-Hawk," which seems to simultaneously suggest that the Titans are well-known to the general public and decidedly obscure, depending on what the current subplot requires.
The Titans' inability to maintain a low profile has become something of a running gag with fans of the show and a frequent point of criticism from the show's detractors. Many have pointed out that Dick Grayson frequently made no effort to hide his identity when going undercover, not bothering to use a disguise or an alias. Donna Troy was similarly caviler when it came to using her lasso while in her civilian garb and the odds were 50/50 as to whether or not Hank Hall and Dawn Granger would bother to change into costume before going off on a mission as Hawk and Dove. Perhaps the most egregious example of this was Jason Todd's shouting "Titans are back, bitches!" into a live television camera in the season 2 premiere. (Todd was notably the only member of the team in costume at the time.)
Titans, season 2, episode 12, "Faux-Hawk," tries to have it both ways with two of its storylines. In the first, Rose Wilson confesses to Jason Todd that she was planted among the Titans as a mole by her father, Deathstroke. Rose went on to tell Jason, whom she had become romantically involved with, about how her father had lied to her about the Titans killing her half-brother, Jericho, describing them as "a gang of psychopaths, highly trained." Rose's story makes it clear that she had no idea who the Titans were before she began training with her father and believed them to be a literal gang of killers.
This stands in marked contrast to Hank Hall's storyline in "Faux-Hawk." The episode shows that Hawk has begun fighting in underground cage fights in full costume, being introduced to the crowd as "the undisputed leader of the Titans." Fallen back into his old habits of drinking heavily and abusing illegal narcotics, a drunken Hank sells his costume to a teenager for $200 and a tip on where to buy some cocaine.
When Hank sobers up later and realizes what he's done (after seeing a news report regarding someone in his costume having robbed a laundromat) he goes off in search of the "faux-Hawk." He runs into a Titans fanboy at the laundromat, who swears that the robber was wearing the real Hawk costume and that he thought it was a real shame that Hawk was "fighting in his suit for blow" and it was "no wonder Dove left him." This suggests that the Titans were noteworthy enough as a superhero team for Hawk to be singled out as someone's favorite Titan and for Hawk and Dove to be remembered five years after the team broke up.