Expose (other) user's role to anonymous using views
Ok, I figured it out with help from @Dropa. What I had to do was to build my view in a different way than I initially had it set up.

Initially I had my view built as showing fields from profile, and then I pulled in the user information through a relationship.

What I had to do was to switch it around and create the view showing fields from user. This allowed me to display the user information, including the role info that I am after.
All other desired information (e.g. other information from the profile module) is then pulled in through a relationship.