It sounds more confusing that way =P Also, having to find your 'partner', go to their profile and then start a private chat would take too long. We should pick a map with a wide open area that is near to both teams' spawns, then everyone meet there and just partner up.
Half of each team lines up, so there are two lines facing each other. These will be the killers. Allies take the first turn, killing their 'partner'. Behind the now dead player will be another player who then revives this person. Through the whole game they will only revive this person.
Once revived, the player then kills the person who killed them. A second medic, this time from Allies, revives the dead player. So all four people involved will have one point each. Repeat this for 20 minutes and you will be doing fine.
The only trouble could be ammo, so if someone runs out of ammo, they switch with their reviver partner. But the same routine applies.