MGM Controller providing OJT described a loss of separation event during a descent effort when the conflicting aircraft was on another sectors frequency.

Narrative

Training was in progress with numerous aircraft of various types/speeds into the primary airport. We accepted a hand off from BHM Approach on an aircraft that was southbound; leveling at 10;000 but would be climbing with ZTL Sector 14. We accepted a hand off from ZTL 14 on a BE9L headed eastbound at 11;000 to cross the flight path of the point out. When the BE9L checked on; he asked for lower; in preparation for landing at a satellite airport well ahead. Our LOA with ZTL also requires that we descend aircraft as soon as possible. The trainee issued a clearance to the BE9L to descend; taking into account our requirement to descend traffic out of the center airspace and the pilots request for lower; expecting that the Center would also climb the point-out aircraft. After about a mile; I observed that the BE9L was not descending very effectively and the other aircraft did not appear to be climbing. I told the RADAR Data Controller to call the Center sector and coordinate to improve the situation and I told the BE9L to expedite his descent through 9;000 for crossing traffic and to turn 15 degrees right; the turn limitation authorized in the LOA with ZTL. This only improved the situation a little bit; because the BE9L turned and descended slowly. The ZTL sector had not answered the land line call and apparently the RADAR Data Controller had given up trying to make contact. I turned the BE9L further south for a total of about 90 degrees and got on the line to the Center and announced that I was turning the traffic in their airspace reference the traffic that they were working in my airspace. When they finally answered the line; then they climbed the other traffic. The combined descent of one and the climb of the other and the turn resolved the situation. Recommendation; ZTL should not have switched the BE9L to us at 11;000 until he was past the 10;000 FT traffic that they were working in our airspace. ZTL was apparently too busy to answer landlines to coordinate. We should not have descended our traffic through the 10;000 FT point out; but this was done based on the expectation that the Center would keep the point out climbing. I should have taken firmer action sooner to prevent the situation from developing as far as it did.