It's an argument between me and my SO:
When do you get a emergency round? When all players salmonmeter is full, or when one of the player's meter is full?
The chance is based on the combined values of the meter in the top-right for each player, referred to internally as the "team smell". The meter goes from values 0-5 and increases by 1 each game, and you only have a 100% chance of encountering Cohozuna if all team members have a full gauge.
If one player's meter is full, you have an absolute minimum chance of ~1.5%, but if you have stuck with the same team members for at least a few games, the chance should be somewhat greater than that.