1 Corinthians 12 indicates in a number of passages that the gifts can only be given according to God's will - and His only (I'll cite 3). There are some indications as well that not everyone will have one of these gifts.
Mormonism, as well as some Charismatics and WoF teachers (all based on my experience with both FLDS/LDS as well as frequent interaction with Charismatics/WoF), teaches these gifts can be imparted with the laying of hands or made manifest through baptism. Important to note, however, that a very large majority of these focus on the speaking of tongues, and (from a Linguists perspective) many do not.
1 Cor 12:11 (Concerning Spiritual Gifts)
"All these are empowered by one and the same Spirit, who apportions to each one individually has He wills."
"But as it is, God arranged the members in the body, each one of them, as He chose."
Now you are the body of Christ and individually members of it. And God has appointed in the church first apostles, second prophets, third teachers, then miracles, then gifts of healing, helping, administrating, and various kinds of tongues.
The Apostles, yes, were given the gift to pass on the Spiritual Gifts - and this, too, was a Gift from the Lord Himself:
Acts 8 we see a man try to use money to pay the Apostles to lay there hands on him, to which Peter replies, in verse 20:
But Peter said unto him, "Thy money perish with thee, because thou hast thought that the gift of God may be purchased with money."
We see, through Scripture, that the gifts can only be made manifest by God's will. The Apostles, under very special command (Jesus, the Lord in the flesh), were able to impart these gifts by laying their hands on people.