Hi,
Central locking isn’t the same as remote locking in MX-5 specs. Central locking should unlock the passenger door when you unlock the driver’s door (with the key).
From the sound of it, you don’t have central locking then. 
(If you did have central locking, but not remote locking, it is possible to retrofit the OEM remote module as described here - I have done so, and it works.)
The immobiliser works by detecting a small transponder built into the key.