I think in such circumstances they would share the costs equally.
I think it would be extremely unlikely for you to get compensation from both parties as you would have to show a letter of confirmation from the airline that your flight was cancelled
So, Romulan, your home is robbed ,or scaffolding falls on your car, or you slip on a puddle in the supermarket - but in each case the person liable compensates you for your loss/damage/injury .
Yes I imagine the issue will be in the small print of the insurance policy you purchased which will state that any compensation received from other sources (airline etc.) will be deductable from any insurance payout.
In my experience on claim forms they always ask a question about payments received from airlines etc. - answering this untruthfully would be insurance fraud I believe