'Motor' actually has two parts - motor and switch.
What is happening here is that the switch is working (since when you unlock driver's side with key, the switch signals the rest of the system to unlock), but the motor probably isn't working, so that when unlocking from the passenger's side, the driver's motor is supposed to unlock, it doesn't, and then the switch is still in the 'lock' position on one side, so all the doors lock again.
Doing the swap side-to-side (as Transcedental suggests) will elminate broken wiring as a possible cause for the motor not to function.
Quite common is to hear the motor going, but for there to be no movement of the lock - a piece of sliding plastic breaks inside and can't be repaired easily.
The most economic solution is to attach a standard central lock motor (available individually or as part of a kit), which can replace the motor part of the standard latch assembly while you still leave the switch connected and working. The standard motor can be fastened to the door subframe, with a rod link to the existing lock pin. You only need the two-wire type, which you connect to the existing motor wires (I can't remember which colours off-hand, but PM me if you need me to look them up).
Standard new central-lock motor is worth about four pounds in your money

A replacement latch from another Tipo would be about twenty pounds over here, maybe more. As for the new parts price, I dread to think...
Thanks!
-Alex