I recently purchased a 1999 a3 and have realised that the passenger side light is permanently on. Is this normal or a problem?

If I switch on the sidelights then both come on, switch off and just the drivers side goes off.

No, it is not normal. Both sides should work the same. By "permanently on" do you mean it is lit even when the motor is off and key removed? Or it's just always on when key is inserted and turned? Either way further investigation is needed.

When the ignition is switched on the single side light turns on. I also noticed that the dash is lit up all the time the ignition is on too, not just when the side lights are turned on. Any ideas what I could check?

Can you specify which light you are referring to? When I first read this I was thinking your were asking about the exterior "passenger side light", the side mounted turn signal. Now I'm wondering if you are referring to an interior light. If so which, the front overhead console or the rear seat overhead?
My dash back lighting for the gauges flashes on when the key is turned but quickly goes out. Comes back on when the parking/head lights are turned on.
edit: have no idea how much the 99 & 06 have in common in regards to the lighting but wouldn't think they work any differently

It's the exterior headlight. Its the same bulb as when you turn on the parking mode (e.g. Stopped and the indicator stalk pushed down).

OK, since there is voltage applied to the bulb when it shouldn't be, I think you should start at the headlight switch. Then divide & troubleshoot the wire along it's path to the headlight. Could be as simple as the switch itself so you may want to disconnect it first and check the headlight before digging into the wiring any further.