In the question How is one supposed to "know" whether or not the Book of Mormon is true?, the accepted answer affirms:
But the witness of the Holy Ghost is testimony from God's spirit directly to yours. It's a very personal affair, and it didn't happen to someone else. It happened to you, and you know what you received from the Lord, and no alternate interpretation from some other person who hasn't experienced it can change the reality of that witness. Simply put, it's the only truly trustworthy evidence there is, because it comes directly from the only perfectly trustworthy source there is.
As to how one is supposed to recognize the answer, this will probably sound like a cop-out, but the best answer is "you'll know it when it happens." The problem is that any description requires a common frame of reference, and the witness of the Holy Ghost is a unique experience that isn't like anything that would be familiar to anyone who doesn't already have experience with it. It's often described as a strong feeling of peace, and a sensation of warmth, a "burning in your bosom" to use the scriptural language, but it's not the same as physical heat. But once one has experienced it, they truly know that they have received a testimony from the Lord.
Seconding the answer above, mormonwiki affirms the following:
One of the foundations of Mormonism is its insistence that a person seek the truth by praying for a private, special revelation from the Holy Spirit 
"In answer to our prayers, the Holy Ghost will teach us through our feelings and thoughts... Heavenly Father will answer their prayers, typically through feelings of their hearts and thoughts in their minds." (Preach the Gospel, p. 39; this is the "2004 handbook utilized by the Mormon missionaries") 
What they receive is sometimes called a burning in the bosom as a confirmation of truth. Mormons frequently appeal to James 1:5 for this, especially given that their founder, Joseph Smith, claimed that this was the verse and method he used for finding the truth. This is often accompanied by the insistence that one suspend judgment of his or her religion (even in the face of its historical and theological problems) until he or she has read the Book of Mormon and received, by prayer, a special revelation from the Holy Spirit of its truthfulness.
(*) All emphasis mine.
Question: Do other Christian denominations/groups have an equivalent concept to Mormonism's burning in the bosom, or at least acknowledge that similar experiences are legitimate and truly happen from time to time?