There are a number of things that attribute to the desire of youth to become missionaries in the Mormon church. For those of us that were raised in the church, we were taught how wonderful it was going to be to go on a mission. Our grandparents, parents, cousins, friends, etc. all talked about how great their missions were and intrigued us on how neat it would be to serve half way across the country or world even.
We were given saving boxes in church that had three slots in it. One for Tithing, one for Savings, and one for Missions. Typically you were to put 10% of your allowance, etc. into each section. I have to admit, I was never very good at saving money so my box was usually pretty empty. In order for me to go on my mission, my dad paid the cost of my mission. Others pay for their own out of their savings. Even others are sponsored by individuals in the church who are willing to help those who can't afford the costs.
At the time I served my mission, everyone paid the same amount into the mission fund every month (I believe it was roughly $325ish). That money was then placed in a mission fund and divvied out among all missionaries around the world based on costs of living. While I was in Russia, my monthly allowance was $200 (roughly 1,000,000 rubles at the time I was there). Our travel and rent also came out of this general missionary fund.
This way we could easily say that there were no paid missionaries in the Mormon church. It became a uniqueness that we were proud of. Again, work righteousness we were taking care of our own way and not relying on the church to provide for us (in most cases).
Another bonus was that we knew we would not be called to serve in the state where we lived. I remember when I opened my "call" letter. I did not know where I was going to be sent. No missionary in the Mormon church selects where they will serve. It is chosen for you at the First Presidency level (so we were told, not sure how accurate it is...). Anyway, my brother was sitting at the table when I was opening and trying to read my letter. I was so excited to learn where I was going.
When I found where it stated that I would be serving in Russia, I started screaming at the top of my lungs. My poor brother had to cover his ears I was so excited. In fact, I think my entire apartment building heard me screaming. The reason I am sharing this is to help you understand how important these "calls" are to Mormons. A mission ranks right up there with a wedding (for a girl).
"According to the grace of God which was given to me, like a wise master builder I laid a foundation, and another is building on it. But each man must be careful how he builds on it. For no man can lay a foundation other than the one which is laid, which is Jesus Christ."
1 Corinthians 3:10-11