I play both to sing to, plus (recently) a cittern too. A lot of the songs I do I learn for more than one instrument, so I can sing them even if I haven't got the concertina with me etc.

I tend to find that accompanying with 'tina I tend to play the tune, then add chords and runs as appropriate; with the guitar it's chords and runs with elements of the tune or countermelody to taste. Strangely, sometimes the two end up very similar. The stuff I'd tend to accompany with just strumming and melody lines over ringing open strings or drones I find work much better on cittern now.

I'd like the opportunity to record myself with both concertina and guitar to see how well they fit together.