When Paul was introducing the subject of Spiritual Gifts in I Corinthians, he made this statement: "To each is given the manifestation of the Spirit FOR THE COMMON GOOD." (I Cor. 12:7; emphasis mine) This was a basic parameter for the entire discussion that was to follow.
Later, he writes: "He who speaks in a tongue edifies himself..." (I Cor. 14:4a) Those who believe in the 'private prayer language' brand of tongues interpret this 'edification' as a good thing, but Paul meant precisely the opposite. The use of an alleged manifestation of the Spirit to edify one's self directly violates one of the basic purposes for the existence of those Gifts, according to Paul's earlier statement. True manifestations of the Holy Spirit used correctly ALWAYS benefit someone else, not ourself.
People who speak in 'private prayer language' tongues are 'edified' by the false belief that their babbling is evidence of the presence of the Holy Spirit in their mind and heart. This makes them feel good and 'special,' and thus they are edified. However, according to Paul, true Spiritual Gifts are meant to edify OTHERS, and in the context of Spiritual Gifts self-edification is a BAD thing, and evidence that what's going on is NOT of God.
The true Gift of Tongues was the ability to speak in a language unknown to the speaker but known to at least one person present, for the purpose of edifying THAT PERSON OR PEOPLE, NOT THE SPEAKER! The only way to justify the 'private prayer language' version of tongues is to separate this one gift from the basic rules which govern the use of all the others. There is no Scriptural basis for doing this.