But that cannot deny that in our heart of hearts we believe the other to be wrong. Most Christians and Muslims believe the other group to be doomed to God's punishment.
This is very true I think. I wouldn't call it a 'competition'. It is more trying to let people know of our faith and inshaAllah (God wiling) showing them that it is the truth. As Muslims, by doing, this we are pleasing Allah (subhana wa ta'ala) and doing something beneficial for other people.
It begs the question, whether we are friends despite thinking that, or because of it (i.e. feeling pity for the other).
It also begs the question, whether we are friends just for the sake of being friends, or whether deep down we are still hoping to convert the other?
On this matter, I don't think you can make friends with people from other faiths JUST to convert them. The fact is that we enjoy being with them and get along, otherwise we would not be friends at all and would just be arguing all the time.
We care about our friends and want what is best for them which is why I think that deep down we would love for them to convert but we wouldn't force our religion on them so as to break that friendship because what would be the point in that? That would deter them away from the religion as we would be showing an awful view of the religion and causing rifts between religions which is stupid and would not be pleasing Allah (swt).
As Allah(swt) says in the Qur'an:
There is no compulsion in religion [2:256]
Obviously this is my opinion. What do other people think?