Translation:I trust neither their president nor ours.
Since I'm not a grammarian, this isn't exactly a scholarly explanation but maybe it will be helpful.
In the given sentence --"I trust neither their president nor ours" -- the verb is the same in both parts of the sentence and it is applied to both people. It is as if you were saying, "I don't trust their president and I don't trust ours either." Here you are saying one thing about two people.
In "I neither trust their president nor ours," "neither" is incorrectly placed for the intended meaning. With "neither" placed before the verb, another verb is expected in the other half of the sentence -- for example, "I neither trust their president nor believe what he says." Here you are saying two things about one person.