Not big on my info on the trade union movement, what happens in an average workplace if two colleagues in a dispute with each other are represented by the same union.
First port of call is to try and sort the dispute out between them, with the union acting as a moderator/go between. The last thing they want is either party going in the book as being accused of something, or being seen to throw accusations around willy nilly.
If it has to go through formal channels such as a grievance process, then yes both will get a rep from the union (they can always call someone in from out of branch as a neutral if needs), but the likelyhood is the reps will always be looking to bring the two parties to a mutual solution based on understanding each sides viewpoint.
If it is a case of where someone is wrong, the rep should be trying to explain where they are wrong, and what company policy/anti-discrimination legislation they have broken, and explain the situation to them. If they are too stubborn to admit defeat, then the rep just has to do the best they can for the person, whilst managing their expectations of sucess.
Actually a mate of mine over at Unison was telling me of an old case he had once where a bloke lamped his colleague as he found he'd been banging his wife behind his back. Was a case of inform the management of the mitigating circs, so the guy who threw hands got off pretty lightly, and look for ways the attacked could continue to work for the company in another department, away from any further ambushes.
I dont know of any cases in our branch in the last year thats been member v member going through a formal grievance. Mainly its member v company action (though the manager acting on behalf of the company is invarably a member, they tend to invoke HR/senior management to fight their corner, as its usually their guidance thats caused the issue!)