I always like people to sing harmonies on refrains and choruses, but I don't think it's appropriate to sing harmony (or to sing at all) on the verses unless it is a very well known song where lots of people know not just the verses, but the same version as the singer.

It's similar to the situation where people try to back up a singer with an instrument. They may not have the same timing as the singer, or the same phrasing and feeling, and they just interfere with what the singer is doing. Sometimes they throw the singer off complete, sometimes it just turns into a different type of song.

It depends a lot on context or the specific song and venue. You just have to be sensitive. I remember once I started to sing Janie Voss' "Standing Behind Some Man." Songster Bob on guitar, Pauline L on fiddle, and various others started backing me up. It is in straight 3/4 time and has straightforward chords, it isn't a ballad that needs expressive timing and phrasing and therefore might be unsuitable for accompaniment, and it was a great jam. What a good time. That still stands out in my memory, especially Pauline's fiddle line.

And I have friends who know the accompaniment to some of my favorite songs and I'd be disappointed if they didn't join in. Other times, I've occasionally asked someone not to accompany me because it just didn't fit.

I suspect harmony is almost a complete parallel. Also, there are some singers who have trouble holding the tune if someone sings a harmony.