The tendency may have started out after Bernardo Marconi - decayed nobleman - got so much in debt after playing cards (the Ridotto?) and 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, and anonymous doing this way.
And 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 the bauta mask was frequently used, nevertheless, to get some money and from other persons to attain personal - not always legal - purposes: spies, “sbirri” (the cops of those times), killers, pimps and prostitutes used it too.