Casey, I've been thinking of your question for almost ten days. As Benny says, students learn proper English. Troglodyte writes that you can change the way of speaking by its level. I think, Casey, you knew about that.
I'll write my thought along with your three lines. I hope my comment will be useful.
What you are questioning;
> Is that just another part of our personality or does that just mimic part of our already existing traits?
I recalled that I was told that I, my personality, sounded nicer in Japanese than when I wrote in English. My English level might affect that, but still, this fact points out that the impression of people are changeable by its language.
Adding on it, you are asking;
>Do you talk differently to different people?
Advanced level learners can do it as Troglodyte writes, but again, thinking of your first question that whether it happens in speaking another language, I would say that another language makes your different perspective and talk a different way to different people. It doesn't always depend on speaking level, beginner or advanced. Sometimes, do. Other times don't.
That's why you are saying;
>I hear people say we get a new view on things when we learn a new language and that we get a new personality that goes with that.
I think that learning another language guides you to a new perspective. Usually, people who speak multiple languages have various thoughts, so they can talk differently and more effective. That's why people get together on this forum to hear new and interesting ideas and opinions.
And my question is that whether people on this forum show their real personalities by its writing?
For example, when I exchange private messages with them, some of them show a more relaxed attitude that makes me comfortable. This is what Casey says: talk differently to different people, that is the difference between talking to many people and individuals here. I think they have both style and can use to fit the situation. I call them "smart".