Confidingly is how someone speaks or acts when they want to tell you something private. They often lower their voice, lean closer, and talk as if the words are for you alone. It describes the manner, not the secret itself: she spoke confidingly even if the news was small. Like confiding, it sounds a little literary. In casual talk people more often say she told me in private or she confided in me. You still hear it when a writer wants you to see the intimacy of the moment.