I am a WELS Lutheran. The idea that we refuse to pray with other Christians is a carricature of us. It is not totally accurate. My pastor says that public prayer is always off limits but private prayer is something that requires a bit of discretion.
I have a good example of something that happened in our church. The parocial school that is attached to our church was getting ready to play a basketball game against a Baptist school. The Baptist school asked us if we wanted to have a prayer before the game started. We had to decline. It is not that we think that the Baptist Church is not a Christian church. It is that we are concerned that we would be giving a false appearance of unity with the Baptists (or whatever other church we are referring to). There are doctrinal differences between Baptists and WELS. We have great respect for our christian brethren in the Baptist church but we feel that they error in many different ways. We feel that by praying with them we are sending them a signal that we are in doctrinal agreement with them.
Private prayers are a bit different. Each situation has to be discerned. My pastor says that if I am inviting my friend to my house and we are going to have a meal in my house that I should say to my friend "I like to pray before my meals. If you want you are welcome to join me or if you do not want to you are not required to". We leave it up to the individual. In that situation I am in control of the situation. It is my house and it is my prayer so I am controlling the message that goes to God. Now if I go to my friend's house and my friend wants to pray then I am told that I should sit there quietly while my friend says his prayer. I am to be very respectful and not cause a scene as I have a great deal of respect for my friend. I just happen to have some differences with him. It is an act of love because I am letting my friend know that he is in error in his ways.
We do not believe that other church bodies are not Christian or that we are better than anyone else. We just believe that their beliefs are different and deserving of the same respect that ours are. Like if a Charismatic Christian is praying I do not think that my Charismatic friend that is a Christian would appreciate his or her prayer to God being dilluted by a person like me that hold a cessationist theological position (this is an example). My Charismatic friend has a right to have a pure prayer sent to God with a clear signal of what he wishes to pray to God for. It would be very disrectful on my part to pray with him or her and dillute his or her message to God.
There are situations where this does not apply. Like if I am praying for a family member that is not WELS who is sick and dying with some other non WELS family members. Or like say if I am praying with someone that is hearing about Christ for the first time and is interested in learning more. In those situations my praying with them would not be dilluting the message or giving a false impression of doctrinal unity.
As WELS Lutherans we are told to make our own judgment in each and every situation as to if it is appropriate or not to pray with someone else. Not each and every WELS member is going to agree on each and every situation each and every time. There are differences in practice but that is OK. I have seen some WELS pastors that will say that it is OK to make the dinner prayer in every situation as they do not believe that blessing a meal is showing doctrinal agreement. They disagree with some others but that is ok. We do have the general concept in line.