TRACON Controller providing OJT described an airspace incursion event; distracted by side conversations and then failing to note proper read back hear back coordination.

Narrative

RADAR training [was] in progress. Trainee certified on Final RADAR and close to being certified on East RADAR. One aircraft involved. Another Approach owns surface to 10;000. A BE20 was inbound to the airport from the east and was at 11;000 in Centers airspace; above the neighboring Approach's airspace. The BE20 was over the VOR. Trainee called the neighboring Approach and requested a point out for the BE20 in order to descend him. Trainer did not notice coordination being done due to conversation with another trainee. Neighboring Controller stated 'Unable; call me back when the BE20 is ten miles west of our VOR.' Trainee misunderstood the other Controller and instructed the BE20 to descend to 5;000. Trainer noticed the BE20 descending and asked trainee if the BE20 was pointed out; to which trainee replied 'affirmative.' Minutes later; the Neighboring Controller called to advise that the BE20 was not pointed out with them and that he had instructed trainee to call back in order to get a point out for the BE20. The Controller then stated that the BE20 was now pointed out with them; and the aircraft continued descending to 5;000. [One should] not allow side conversations to distract from proper hear back; read back.