Basically, they all get together and put their pets in a "pot" of pets to trade and call it a "Shuffle." There are different versions of this "game." (The petpage ~medicare explains the rules the users have dreamed up.)
In this version of the game, regardless of what pet is offered on the pet that is up for trade, whether it is good or bad, the original pet trader can decide to pass on every offer and has no obligation to trade their pet.
This version is therefore very low risk, but accordingly, it is very low (usually no) reward as well.
A lot of times, a Pass Shuffle is people trotting out pets they just want to show off and often had no intention of trading anyway, just to see what kind of offers they could get.