i think they are not called terrible twos for no reason.. haha.. my girl is the same, just that she does not beat people. She used to roll on the floor and cried loudly but i have got used to other ppl's stares by just standing aside and ignoring her. There are many parenting books and tips available but it really depends on the child. My hubby always says i have theories but when it comes to actual discipline, my girl is more afraid of him.
But i think it is true that when she throws a tantrum, say she wants to buy something, we can't always give in coz they seem to be testing their boundaries and will kick a bigger tantrum the next time until we give in. Some things i tried include distracting her with other things, telling her we are not going to buy anything when we go into a shop, praising her when she is well behaved etc.
I do hit her palms when she does naughty things and i will tell her i do not like this kind of behaviour and asked her if she wants others to do the same to her. But since your mum is the main caregiver, she must have the same expectations as you. Think children are very smart these days, they know who are the ones they can 'bully'.
I read the super nanny books and she mentioned that disciplining should start only after two but think we have to start early with our tones of speaking. Low and firm tone and speak to them at their eye level. Keep trying.. (btw, my skin has got thicker so i m not that embarrassed anymore when people are staring)