Simple - you know they're a prophet when you feel the Spirit around them.
(Guide to the Scriptures | P Prophet.:Entry)
PROPHET. See also President; Prophecy, Prophesy; Revelation; Seer
A person who has been called by and speaks for God. As a messenger of God, a prophet receives commandments, prophecies, and revelations from God. His responsibility is to make known God's will and true character to mankind and to show the meaning of his dealings with them. A prophet denounces sin and foretells its consequences. He is a preacher of righteousness. On occasion, prophets may be inspired to foretell the future for the benefit of mankind. His primary responsibility, however, is to bear witness of Christ. The President of The Church of Jesus Christ of Latter-day Saints is God's prophet on earth today. Members of the First Presidency and the Twelve Apostles are sustained as prophets, seers, and revelators.
Would God that all the Lord's people were prophets, Num. 11:29. If there be a prophet, I the Lord will make myself known unto him in a vision, Num. 12:6. The Lord testified against Israel by all the prophets, 2 Kgs. 17:13 (2 Chr. 36:15–16; Jer. 7:25). I ordained thee a prophet unto the nations, Jer. 1:5, 7. He revealeth his secret unto his servants the prophets, Amos 3:7. He spake by the mouth of his holy prophets, Luke 1:70 (Acts 3:21). All the prophets witnessed of Christ, Acts 10:43. God hath set prophets in the Church, 1 Cor. 12:28 (Eph. 4:11). The Church is built on a foundation of Apostles and prophets, Eph. 2:19–20. The people have rejected the words of the prophets, 1 Ne. 3:17–18 (2 Ne. 26:3). By the Spirit are all things made known unto the prophets, 1 Ne. 22:1–2. Christ came to the Nephites to fulfill all he had spoken by the mouths of his holy prophets, 3 Ne. 1:13 (D&C 42:39). Those who will not give heed to the words of prophets shall be cut off, D&C 1:14. Those who believe the words of the prophets have eternal life, D&C 20:26. The prophet's word ye shall receive, as if from mine own mouth, D&C 21:4–6. Revelations and commandments for the Church come only through the one appointed, D&C 43:1–7. The duty of the President is to preside over the whole Church and to be like unto Moses, to be prophet, D&C 107:91–92. We believe in prophets, A of F 1:6.