It happens all the time. We absolutely believe in the power of prayer. I can't tell you how many times throughout my life that I have received answers to my prayers. 
 
As far as your initial question - I think almost all Mormon missionaries have stories that they can tell where this happened to them. It happened to me a couple of times on my mission (which was a couple of decades ago). I know of an older lady in our congregation (who passed away about a year ago) who told me directly that one day several years ago she was in her house in tears because she said she was so lonely. She asked God for help and within 15 minutes 2 mormon missionaries knocked on her door. 
 
Answers to prayers happen all the time - because God loves us. This is especially true for those seeking truth.

It seems like this is a really common occurrence - all my life I've heard first and second hand accounts of this happening. I remember it happening a lot when I was serving as a missionary. In one case we called at someone's house and they came to the door and said they had been praying right then for God to send someone to teach them and when they heard us call, they knew their prayer had been answered. 
 
A cynic would tell you it's all just a bunch of coincidences, despite how often it happens. I look at it as evidence that God cares about each person individually - that he really does listen to and answer prayers.  
 
It's also really humbling to be allowed to be involved in some small way - I know that as a missionary I was just doing what I supposed to do, but that I'm very imperfect and not exactly the greatest "tool" in the Lord's hands, so it's thrilling to know that he can find us useful to help anyway.