as someone who has been accompanying himself on the English concertina since 1976,and someone who has produced two tutors which include some twenty song accompaniments,I would say this:onthe english concertina there are several approaches to accompaniment which can be successful.
1.single line harmony.
2,chords [which may or may not involve single line harmony runs].
3.melody line[putting early or delaying melody notes off or on the beat,and adding harmony so you have occasional two or thee voices]
4.melody with full accompaniment on the off beat,to get something similiar to a duet style[see my recordings of Ball of Yarn]
Finally Ihave avideo on youtube dickmilesmusic
Barbara Allen,Sallyfree and Easy,OnOne AprilMorning,these illustrate different approaches to song accompaniment.
I would advise listening to Killen,Tony Rose,Steve Turner,Keith Kendrick.http://www.dickmiles.com