I find it difficult to give a definite answer to this.
You can put between in front of either or but you don't have to.
I can choose either red or green.
I can choose between either red or green.
I can choose red or green.
I can choose between red and green.
To my ears, you sentence sounds a little better with "between"
1)I need to choose between either studying abroad or staying in my home country .
But I have no formal grammar training, I am just a native speaker from the UK.