You have resistance in the circuit. Your mission, if you choose to accept it, is to determine if the resistance is on the Power side of the circuit or the Ground side of the circuit.

Since the passenger side visor light works I'd suspect the switch on the driver's side visor OR the bulb sockets is the problem (check for corrosion or a bad connection). That's because both visors share the same Power and Ground points.

Let's have a look-see as to how these circuits are wired up.

I highlighted the Power side with Red and the Ground side with Green.

Here's the wiring connector view of the driver side visor light. 

 If you want to troubleshoot this, you could attach alligator clips and a short piece of wire to both ends of the visor bulb and stick each wire into the (disconnected) wiring connector terminals. If it lights brightly then it's the visor light switch or a faulty bulb socket connection.

If it doesn't light brightly you can leave the wire in terminal 1 (Power) and provide the other wire with a good Ground. If it lights brightly then the issue is on the Ground side (refer to the wiring diagram).

If it doesn't light brightly you can leave the wire in terminal 2 (Ground) and provide the other wire with Power. If it lights brightly then the issue is on the Power side (refer to the wiring diagram).

Or you can wait until you're my age and appreciate dimly lighted mirror views

Check the wiring, it may have deteriorated after 14 years of flexing.

Are you referring to the wiring on to the visor? Other sites?

Many thanks, all. Will give your suggestions a try.