This is a normal case of battle of the wills. The ages of your children is about where most parents never want to repeat...thus the term terrible twos.
It's important to establish who's boss at this stage. Children develop their personalities and life-long habits in the first 3 years, so it's important to let your children know that you are the parent. If you don't establish your position in their lives now...it will be harder on you as they grow older.
My girls never had tantrums, but I worked in the 1 - 2 year olds in my church (was I crazy?), so I was exposed to a myriad of personality formations :)
Earlier last fall (2007), one little 2 year old boy had an issue with not only screaming, but biting and pinching as well. It was so bad one morning, that I literally held him tight...holding on to his hands and making sure he couldn't bite me. He struggled, but I held on strong. I didn't allow his screaming and squirming to lighten my hold. After about 2 hours, he calmed down and fell asleep.
Today, I am his best friend! Ever since that day, he comes up to me, sits on my lap and has never had a tantrum with me. The funny thing is that he still has tantrums with his mother...but as soon as he sees me, he stops and runs over to me and wants to sit on my lap.
I believe children want to know where the line is...they find security in the person who breaks their will.
I've heard the saying..."if you don't break their will...it will break you"
Hope this helps...