Unless things have changed substantially, all divers must have a current signed medical to dive in Malta. A brief exam can be arranged through physicians affiliated with many dive centers. When I was in Malta for the summer of 2001, the medical exam cost Lm3 (about US $6.75) using the shop's physician. Your cost may vary. As it was explained to me, you can go with an on-island physician or bring a clearance from home.
To dive without a guide, you must procure a Maltese Sport Diving License. The shop I dealt with absolutely would not rent tanks without both an internationally recognized diving certification card and the Maltese Sport Diving License.
To procure the Maltese Sport DIving License I had to submit the application fee (about Lm2 or Lm3, don't recall), a copy of my diving medical signed by the doctor, and a copy of proof of certification to PADI AOW or higher (or other comparable rating according to certifying agency). The dive shop I worked with handled this application for me. As I understand it, the information is sent to be filed with the hyperbaric facility at Luqa hospital.
Of course, I encourage the reader to contact a dive center in your travel destination for current information about local regulations.