Real Vocalist Please Answer! How do you learn to harmonize with someone?

So I've been singing all my life not professionally, I've never had any lessons so I have no professional background knowledge in this at all. I thought I knew how to harmonize with people because when I sing with audio it sounds fine, or I think I'm doing pretty well. But after watching the voice this season with contestants Whitney and Shannon, and hearing how they harmonize and how they're applauded for it, I realize that I'm not really harmonizing. I mean they aren't the first for me to notice, I just always thought people who harmonized like them were doing it wrong. But I was obviously wrong. Heres a background on how I sing: For example if I were singing w/someone with a lower register I would higher mines not too high just to the right amount. When I read the how toos or videos that show how to harmonize that's what they say to do, so I'm confused. Bc what I notice is that real harmonizing is like singing two different songs. One voice sings the song one way while the other sings it another but together in registers that compliment one another. I hope this makes sense. Let me try again, like for example if I were singing in a duo, the other person would sing the song the way the artist sings it, then I would sing it in a totally different way basically a new rendition but one that compliments the real way to sing it. I just don't understand how it works. Is that what I'm suppose to do completely sing a new rendition of the song? Please do help! Thank you!!