What does it mean to be a person who speaks with authority?

Speaking with "authority" does not necessarily mean that you have "authorization" to say something. Anyone, can say anything, anytime. Lawmen speak in tones that command attention, people talking to other people trying to impose their will on them can speak with those voices. People don't have to listen to it, and certainly can voice their disapproval. HowEVER, a person who speaks with authority is more or less speaking with the blessing of the Holy Spirit. The Holy Spirit gives the person boldness to speak the things concerning Christ and His church. The boldness to speak with confidence, to reach the hearts of those who are willing to listen. And those who will hear, will recognize the Spirit in him, as Christ recognizes His own. As Christ said: "My sheep hear my voice, and I know them." And to him that speaks with authority, his Words will be sharp.