You have the right of way if you are turning left from a through road; traffic coming towards you must give way. In New Zealand it used to be the other way around, but the law was changed in 2012.
The only time you wouldn't have priority when turning left if there's another driver turning right is if you are on a single lane roundabout and the driver turning right is coming from the right. You would have to wait for that vehicle to pass.
If you are both turning into a one-way street and the street has two lanes then you can both turn at the same time. This is because you will turn into the left lane and the other vehicle will turn into the right lane.
You must stop and wait for pedestrians already crossing the road, though, as will the vehicle turning right.