As an administrator as well as an user. As soon as I upload an avatar and as soon as its icon is displayed in the upper right corner the browser complains about the mixed secure/unsecure content on all pages served with HTTPS (e.g. edit profile page etc.). If I check the html the uploaded avatar images is linked to as HTTP instead of HTTPS).
But if this is not a known common issue or not easily reproducable it might as well be related to my theme.