As far as India is concerned, wearing scarf is not that a threat; but, it surely conceals the identity of the person if worn over face. As face is the primary part with which a person can be identified, face should be let free when entered into private property (Malls, Jewelry outlets, Apparels outlets ects) or indoor of the public property (ATM - anywhere it is banned to conceal your face in ATM, Bank, Antic pieces Museum etc).
In response to Ms Juana, it would not make any difference if any attire (loose clothe) or a thing (dickies or backpacks) could conceal stuffs until a person is identified by his face. If the face itself is concealed as in case of burglary where thieves and robbers use it like a tool to escape law (and may escaped so far), it is advised not to cover their face at least when indoors or entering a restricted property. Helmets are made legal while on driving but in case of scarfs, it is not as if so.
I do know, girls use scarfs to avoid allergy and direct exposure to sun, but need to avoid when entered reserved properties for security reasons.
And please don't blame me if I say, I have few friends (girls) in my circle who scarf their face to conceal identity when roaming with their boyfriends. Lol. No offense though :)
"Whatever may be the situation of the forest,
a Lion would never eat Grass!"