ZMA Controller described an airspace incursion event when EYW controllers failed to initiate a DM message activating the data block. The reporter indicated this oversight is a frequent occurrence.

Narrative

I was training on the MTH RADAR and noticed a Mode C intruder traveling approximately 280 KTS at an IFR altitude. I questioned my trainee if he had looked into the aircraft or was aware of who it is. I had my D-Side call EYW Approach to inquire about the beacon code. After they looked into it; EYW gave us the call sign and stated they thought he is IFR. Meanwhile; we had descending jet traffic to 10;000 FT in conflict landing MIA. After we became aware that the TBM7 was IFR; we amended our descending traffic's altitude. Had we not done this; there would have been two IFR aircraft with no approved separation. In addition; EYW Approach had not switched communication to us regardless of having either a track or RADAR contact. There have been many aircraft departing EYW Approach who have not been DM'ed; which require additional steps for the Center to gain track control and could end up in a beacon code change. Whoever is clearing the aircraft for release should have to manually enter a departure message.