My next door neighbour has been taking out the ivy from his privet hedge and it has created huge holes - I live with mine as it keeps the hedge thick and the birds like nesting in it. I think it would be difficult not to kill off the hedge if you started spraying it with weedkiller.

If you can get into the base of the hedge then you could cut through the stems of the ivy with a pair of loppers.

I agree with Bulbaholic. The trouble with ivy is that it excludes light to the host plant and while it might be bare for a short time privet should come away again. I would put Ivy on the thug register if there is such a thing. Trees, hedges and walls all over Britain are suffering from this hostile predator. It only needs a chink in a wall to send out a root and before you know it the root is big enough to force the stones apart. Whatever happened to Englands green and pleasant land. Monotony will rule.

I agree with Bulbaholic too. Ivy is NOT that difficult to remove, especially if you've had to wrestle with bindweed taking over a hedge! Our cottage and barn here was absolutely covered in ivy when we bought it, with stems like tree trunks. After a week or two with secateurs, loppers and saws we removed the lot and regular attention prevents any new ivy growing.
It should be a fairly easy matter to snip off the ivy at ground level and keep it from re-growing each year without recourse to chemicals. Ivy is not one of those persistent weeds which cannot be eradicated by simple mechanical means.