We should all have a heart for the lost. We should all be praying for the lost to be saved, those we know and those we don't know. I believe the Holy Spirit gives all true believers a burden to see people saved. After all, Jesus came "to seek and to save that which was lost" (Luke 19:10).
I think a lot of times we just aren't sure how to go about praying for the lost. Some of us believe that human beings have 'free will' and must use that freedom to choose to be saved. So how do we pray for God to save them when God can't save them until they decide to be saved? Some of us believe that God has predetermined the 'elect' from before time. So since we're not sure who's of the elect and whose not, who do we pray for to be saved; If God elected them, they'll be saved and if He didn't no amount of praying will change that. It can certainly be confusing.
I believe that no one in his natural state, that is, in his flesh, ever wants to be saved or have anything to do with God. The Bible tells us that in our flesh, "nothing good dwells" (Rom 7:18). I believe the Bible teaches that sin has corrupted us beyond repair (there remains no ability to want the things of God left in us) and that is why we need to be reborn of the Spirit. I believe that God must give us the will to want him and the ability to desire to do his will. The Bible tells us that it is "God working in you, giving you the desire and the power to do what pleases him" (Phil 2:13). I believe in 'free will', in that we are free to choose what we want. But I also believe that what we want is not (and without God's intervention, will never be) God. The Bible says that we love him because he first loved us. We need God's help to be saved.
I believe we need to pray and ask God to give lost people the desire to be saved, the ability to repent and the will to follow Jesus.