Why do Jehovah's Witnesses not pray with people of other faith? I had it happen several times that Jehovah's Witnesses refused to pray together with me, even though I offered that they say the prayer. The reason, according to them, was that they believe we don't worship the same God.

This happened to me both in France where I was an LDS missionary, and in Germany where I accompanied some.

I understand the reasoning in the case when they would have to participate in a prayer others are saying. But why do they also not want to pray together when given the opportunity to say the prayer themselves?
Both Jehovah's Witnesses and LDS have different meaning of God and Christ, so their understanding the very nature of God differs from the rest of the christians and from each other.

Jesus in Jehovah's Witnesses views is created by God. Whereas in LDS, Jesus is born and is first of many sons. Not to mention that Jehovah for LDS cosmology is Jesus.
See also
What exactly does it mean that Jesus Christ is the son of God?


  When servants of God assemble and pray together, their petitions can be effective. If the group is united in spirit and the prayer offered in their behalf reflects Scriptural principles, God is pleased. JW.org




As for Orthodox views on praying with others - christians can pray by themselves to God in any situation (in mosque for example), but social, common praying cannot be done with heretics. Thats why symbol of faith exists. 


  If anyone pray in company with one who has been excommunicated, he shall be excommunicated himself.
  Canons of the Apostles, nr. 10,45,46


Even catholic and orthodox praying is separate. For example when Alexy II visited France, praying was done in order, not together.


  No one shall join in prayers with heretics or schismatics. Canon rule nr. 33 of the Council of Laodicea


My personal understanding of this is that prayer is a message. Labeling who the target is, is important. You can help with delivery by doing it in group, or through saint, deed or with use of icons. But if its collective, then its important to have same target and message for all participants.