Possession is a specific thing (I think) in football. When the opening OT kickoff is caught by the receiving team, that team has possession. If they fumble and the kicking team recovers, then both teams have had one possession and the game continues in sudden death mode. If the kicking team does an onside kick and recovers it, then they have possession but the receiving team never had possession, so they would still be guaranteed one possession later in the OT.
The team with the first possession attempts to score as normal. Regardless of the outcome of their possession, the other team is guaranteed their own first possession. So if the first team scores, they have to kickoff, otherwise the second team would get the ball through a punt or turnover.
Once the other team gets the ball in one of those ways, then that marks the start of their first possession. Play continues until the end of their first possession. If at that point the score is not tied, then the game is over. If it is tied, the game continues with sudden death.
To recap, if the first team with possession doesn't score, play continues with sudden death. If they do, then the second team is guaranteed at least one possession, and after that play continues with sudden death.
