Did some more tests today on our Jazz. With the power still on, nothing happens when you open the door and get out. When the door is closed there are 4/5 very quiet beeps (lasting 1-2 seconds, if that, in total) which could easily be inaudible in a busy street environment. The beeps seem to come from somewhere near the rear wheels. When you try to lock the car using the fob, nothing happens - the car doesn't lock, the indicator lights don't flash and there is no warning sound i.e. so nothing to alert you.
I have to say I find this very poor for a company like Honda that prides itself on safety features etc. Curiously my CR-V beeps loudly when you open the driver's door with the engine still running, which seems a much better arrangement. I wonder why they didn't stick to that on the Jazz as I would expect it to be a company standard.