Do Mormons secretly pray about you and then consider you included in their church? I met two men on the street who were Mormons. We had a nice discussion and they gave me their book with a bookmark pointing to a verse in it. 

Later that day I met with my friend and he said I shouldn't have accepted anything from them because this is the way how they include you in their church. They pray quietly the prayer of acceptance over you without your knowledge. 

Do Mormons secretly include you in the Church?

How much of this is true? What is the process of one to be accepted in LDS Church?
If they were wearing name tags, they were probably official LDS missionaries. Even if not, Latter-day Saints are allowed to (and in fact, are encouraged to) proselytize informally and refer potential converts to official LDS missionaries. The LDS have a specific set of procedures and practices for seeking out, encouraging, vetting, and accepting new members, which you can read about in the official LDS Preach My Gospel guide. In general, the procedure involves meeting with official LDS missionaries over a period of several weeks of learning, study, and prayer that leads to the receipt of baptism.

Some LDS missions are known for keeping informal notes of contacts made, but you cannot become a "member" of the church without baptism. Accepting materials from missionaries, including the Book of Mormon, tracts, or anything else, cannot make you a Mormon. Only you can make yourself a Mormon by accepting the LDS gospel and receiving baptism by a holder of the LDS priesthood. As to whether or not you should do so, that is solely dependent on your own conscience.