I believe anyone can alter their personality, even to an extreme extent. I would say the majority of people wont 100% get rid of their anxiety. Perhaps they can make it so they dont have a disorder anymore, but they still might have a bit of anxiety or nervousness.
If you really work at it, i think it's possible. I've been working on my social anxiety a lot this year, reading books on brain plasticity, body language, behavior, conversation skills and such, joining social groups, not turning down invitations to go to something (unless I actually am unable to go), working on eye contact and holding a conversation. My friends have noticed a significant improvement, and this is only in a few months.
I've noticed that I enjoy being social more, look forward to social events and meeting people, where I used to dread it.