Abusive people are generally selfish and manipulative. As such, they know that abuse has to be 1:1, with the appearance of normalcy to everyone else. Being rude in public would cede their abusive power ofer any individual.

EXCEPT

When abusive people gain a critical mass of followers in public, the behaviour reserved for 1:1 interactions goes public, and they start behaving in public like they always do in private.

A good example of this is Donald Trump.