It probably started out after Bernardo Marconi, decayed nobleman, who got so much in debt after playing cards (the Ridotto?) that started asking for money in the public street.
Not really an honorable situation, in those times, but the bauta mask surely helped to make him a little more comfortable, doing this.
So we are referred that in 1483, having one's own face covered with a mask while asking for money during the day or night, was highly prohibited.
But it was frequently used, nevertheless, to get some money and from other persons to attain their own purposes: spies, “sbirri” (cops of those times), killers and prostitutes used it too.