Scotland centre Dewey has agreed to join Edinburgh's Magners League rivals Ulster in the summer on the same length of contract.
But Ireland A international Gissing will join the Scottish side from Italian outfit Calvisano.
The 31-year-old previously played for Auckland and Leinster, where he scored 10 tries in 73 games.
Gissing said: "I'm excited about the challenge and looking forward to playing in the Magners League again, as well as being involved in the Heineken Cup.
"I think everything about Edinburgh is great. The club is on the way up and has great facilities.
"They are near the top of the league at the moment and are very ambitions and I'm enthused about playing for a team like that.
"Having played at Leinster for four years, I have been to the city a few times. It's a great place and my wife and I are looking forward to bringing up our baby there."
Edinburgh head coach Lynn Howells said: "Ben is a line-out specialist and an athlete and can cover both the second and back rows.
"At two metres tall and weighing 120kg, he brings a physical presence to the squad.
"When he's faced teams I've coached in the past, he's always caused me problems with his aggressiveness, line-out ability and physicality at the breakdown area."
Dewey's planned exit comes after Gloucester snapped up Edinburgh pair Ally Strokosch and Ally Dickinson.
But executive chairman Alex Carruthers said: "We are continuing to finalise our squad for next season and we shall announce further signings in due course."
Meanwhile, the club have announced a change in its coaching team.
Lynn Howells will now assume responsibility for forward coaching in addition to the overall management of the squad.
Rob Moffat will continue to be responsible for coaching the backs, while Massimo Cuttitta will keep providing specialist scrummaging support.
Iain Paxton, who has been forwards coach since the beginning of the season, will concentrate on player identification, match analysis and community development.
Howells said: "As a forwards coach myself, it has been difficult to be in a position where there were two of us involved in coaching the forwards."