I don't need to know the person I'm praying for. God knows the person I'm praying for, and knew about their situation long before I prayed. God is interested in growing us in our compassion for others and wants us to grow in community. He also uses prayer to draw our attention to His doings in our lives. Does it work? Wow, I'm sad that you have to ask. I've never had a prayer fall on deaf ears.
If I feel moved out of compassion to know more, I ask for more. If I am asked to not pry for details, but just to pray, I will do that as well because again, God knows the whole story.
When I pray, I make is in submission to His will. What I pray more is for people's comfort, understanding, growth in relationship with Him, and for their peace of mind. I have seen enough to know that EVERY bad thing that happens is used for good eventually. God has a miraculous way of turning lemons into proverbial lemonade -- in an amazing, intricate, astounding way. Sometimes it takes years.
You didn't exactly ask, but I dont' believe in a set phrase or posture -- but a moment of mental silence in which all my attention is on God and the person I'm petitioning on behalf of. I honestly think I sound ridiculous when I try to vocalize, so it's usually a silent prayer with (usually) closed eyes. There has been no magical ritual that has worked any better for me than how I currently do it.
The crazy cool thing is that MOST of my prayers are answered within 24 hours -- even if it's just a clear message of "I heard you and you're just going to have to be patient to see how supernaturally and wonderfully I handle this." He seriously blows me away sometimes. I've spontaneously sobbed (happy tears) on account of some of his answers.
The example that comes to mind is when I had an ileostomy several years back. I was a single parent, and I had just started my new job. Problem was, I needed medical supplies in order to be able to work, and I had no money until I got my first paycheck. If my pouch came off at work, I would be too humiliated to ever return. It is so nasty you CANNOT imagine. So I prayed (rummaging through my medicine cabinet), "God, I have no idea how you're going to manage this, but you always do. So let's just agree that I'm going to trust you in this and you're going to use this to grow me in faith." The next two days, when my last pouch should have been dissolving and falling off, it held better than it ever had. No embarrassing situations, no nastiness. When I stopped at the mailbox on the third day, there was a sample pack of ostomy supplies -- exactly what I needed and just enough to hold me until my paycheck. No more, no less. Just enough.
That is just one of THOUSANDS he's answered just the same sort of way. Practical, immediate, and memorable. There is no question, when it happens, where the answer or solution has come from. It never fails to demonstrate his abundant, practical and personal brand of love.
God is SOOOO COOL!!