My wife's got a road bike with drop bars on.

She doesn't get on with the brakes on the drops, so after many months of persisting with them, we're giving in !

We've got a friend who wants to swap my wifes drop 9sp STI Tiagra shifters inc bars for his flat bars plus standard avid brake levers and 9sp deore shifters.

Does anyone know if we will need to swap both mechs too or will we be ok just swapping all handlebar stuff ?

fine swapping just the bar stuff

front mech may be a problem, sometimes.
And road brakes with v-brake levers won't work properly, v-brakes and caliper brakes use different cable pulls.

Front mech won't be a problem if you get a set of road bike flat bar shifters.. Don't know the code I'm afraid! Just don't get MTB ones.. the rear will be ok, but the front all over the place.
Make sure you get brake levers with adjustable pull ratio too. You want to set it low for callipers.

I use 105 10 speed shifter and XTR Shadow 9 speed rear mech (10 speed Ultegra cassette) no problems.