Should I nofollow the social media icons on my own site that link to my own Social Media profiles?*
Do not nofollow your [own] social media profiles. Do not nofollow your own links or links to properties you manage. Only nofollow links if they are low quality sites, you have a commercial relationship with them, or they are competitors.
I can’t see any reason why you would want to nofollow links to your own social networking pages.
*This is different from my Social Media share buttons that are on each Post.