The fact that they were tight is not a good sign. Tight braids hurt hair follicles which could explain the itching. As Darlingdiva said, you could also be allergic to the hair. Synthetic hair makes my scalp itch. It feels as if the hair is prickling me.
For crying out loud, people, why does anyone let someone that is pulling your hair finish braiding at all???
No beauty is worth the pain or risk of ending up like this:
Seriously sistahs, the next time someone's doing your hair, if it feels tight can you please fake a scream and act like you're about to faint so they can stop long enough for you to get out of there? Someone once said that stylists pull so that you can't complain that the braids came out after a week or so. That's BS. You can braid firmly (the rope part of the braid) so that there's no sliding, without braiding TIGHTLY at the base.
People please shop around instead of settling for the first braider you come across. Maybe you'll lose $5 bucks here and there paying the lousy stylist for her time and effort, but don't sit there and take pain just to look cute.
When you find the good braider, it'll have been worth the sacrifice.