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.

No, you shouldn't. Either...or... basically have that "between" already. So, you can say either "I need to choose either studying abroad or staying in my home country." or "I need to choose between studying abroad and staying in my home country." 1.Either...or... 2.Between...and...