Thank you for sharing this with us.
It seems to me it might give some perspective
to some of the other mothers of 3-year-old kids here.
Maybe.
I have not yet read the other answers.
This situation is especially difficult
because you've got so many things all going on at the same time.
Do I understand correctly that DD is the middle of three children?
How old was your older child (DS) when your 3-year-old was born?
And how young is your baby?
Was the 3-year-old in "a better place" before baby arrived?
How is your DS coping
with the ongoing beating he gets from his little sister?
I'm imagining here that she is, at least in part,
reacting extremely intensely to having been displaced by baby.
Up until baby arrived, she was the baby and was coddled
and catered to. Yes?
And then, suddenly . . . . .OOOPS!
You're a big girl now.
I need you to help me, stop being a baby,
I need you to take care of yourself more than you used to.
????
All of which she wasn't nearly ready for.
All of which seems incredibly unfair to her.
All of which she is unable to communicate clearly.
All of which I am projecting/assuming.
Am I even close?
So . . . . there are various things one can do
to help a 3-year-old get through difficulties and inappropriate behavior.
There are various books (Happiest Toddler, et al.)
and I'm sure you'll get all kinds of good advice here.
(and perhaps some bad advice . .. . sighhh).
I'm confident you'll know which is which.
HOWEVER, it seems to me you may not have the energy or patience
to devote the kinds of attention and behavior modification techniques
you will need consistently and lovingly.
I'll read the other answers and see if I want to add anything here.
Sending good thoughts.
===================================
OK. I read the other answers so far.
It seems to me that before you can begin instituting strict discipline and consequences, it would be very very helpful if you can spend some quality time ALONE with her and discuss the situation. Including maybe attempting to draw out her feelings about being displaced. Probably start with how much you miss the special times you used to be able to spend together before baby arrived. Include how proud you are of her good behavior at school. Let her know how much happier everyone at home could be if only she would take some of that behavior she exhibits at school and bring it home with her. Let her know how much you love her.
Then, after you and she have talked about these things --
a dialogue, of course. Not a monologue --
then, you can explain your plans/intentions for helping her learn
not to scream any more.
You'll explain what kinds of warnings you intend to give
and what the consequences will be.
Also in this explanation should be something about
treating her big brother with respect. He has done nothing
to deserve the behavior she has directed at him.
Everyone in your family deserves to be heard, loved, treated with respect.
You can get through to her (I believe) that it is up to her
(as well as everyone else in the family) to get your family to that place, where everyone knows he or she is valued and loved, and where everyone behaves in ways that are loving and respectful to one another.
Let us know, please, how it goes.