Simple harmony is indeed simple - When I found myself trying to sing with two singers who were used to harmomising , we taped all the rehearsals - If I was 'on' someone elses note , we found what ever chord fitted at that pont in the music and I found a note in the chord that wasnt one of theirs ! As easy as that !
Of course , if you want more than simple three part , its a bit more difficult , as the chords get more complicated .