I think you put your finger on it-- two strong and stubborn personalities, and I guess we are close enough that we feel we can snipe at each other's mistakes rather than diplomatically holding our tongues. The instrumental side of things isn't too much of a problem-- he plays mandolin or tenor banjo and I accompany him on guitar. Singing is harder. When we try to give each other "constructive" criticism about phrasing, about harmonies, or just about anything else, we both have the bad habit of taking it as some kind of personal attack-- which usually ends in one or the other of us putting our instrument down and going away. The suggestion may be taken on board later, though-- somewhat sheepishly.
I sing with a band at the moment-- five women and we are all close friends--and we somehow have the knack of working together democratically and with a minimum of arguments.