For the team possession stat, when the game is in progress does one team always have possession or can there ever be a time when neither team is defined as having possession?
For the NHL at least, the term "team possession" can be a little bit of a misnomer considering how it's calculated. It doesn't actually come from somebody holding a stopwatch and keeping track of the time that each time controls the puck. They have tried that method before, but there's the problem that you pose which happens quite often where it's unclear which team has "possession". Instead, there's a couple of statistics called Corsi and Fenwick (or FenClose) (scroll down to Possession Statistics) that have been shown to be good indicators of time of possession, as well as other things. Based on shot attempts, they're sometimes translated to a time, which is the stat that you've probably seen.