The move follows the appointment of a new council chief executive Katherine Kerswell last January, and a subsequent review of activities. This highlighted deficiencies in the council's communications work.
Richard Tapson, assistant to the chief executive, said: 'We have spent the past 12 months talking to staff about the issues facing the council, and communications was one of the things that came up.
'In the past it has been run on a shoestring and we have not had the expertise to portray the things that have happened in a good light', he added.
Tapson said having the new department work across both organisations - believed to be the first arrangement of its kind in local government - would bring greater efficiency.
Under the new arrangement the press and PR officer will primarily look after, and be funded by, the Trust, while the other two appointees will mainly work for, and be funded by, the Council.
Tapson said the money for the new positions had been found by cutting costs elsewhere in the organisation.
Currently the Primary Care Trust has no full time PR staff, while the council has a media officer whose job is to handle all external PR for the authority, which has 8,000 staff.
Tapson said the move was not related to any specific upcoming communications challenges, although the Council's area covers the NEC in Birmingham, and the city's proposed site for a new national stadium - both of which require communications activities. Internal communications within the council is also part of the brief.