I hate barbed wire - evil stuff. First thing we did here was to remove it all.
We planted our hedges from fresh, so slightly different situation. We put a double fence a couple of metres apart, with the hedge down the middle. Cattle on the far side, so had to put in a high extra strand of tensioned wire (on taller stobs) to stop the cattle jumping in (huge cows flicking up their heels and hopping over - quite funny except for what they did to the hedge). But with an established hedge that wouldn't be a problem. If you maintain your hedge by laying then you will need to be able to get at it periodically to do the work, that or remove the fence temporarily. If you are trimming using mechanical means then you don't really need so much space. We have sheep on our side and they keep the sides trimmed to just above the top of the wire (they stand on their back legs and stretch up a surprisingly long way). For the hedges we trim, we work above that line, but those we lay are more of a problem for access.
If your hedge is big and bushy then having the fence up against it wouldn't be a problem - the sides would be trimmed by your animals, the top would stay big and bushy.