Drift diving in Cozumel is awesome and the easiest kind of diving you can do. You consume very little air since you often don't have to kick at all, just relax and enjoy the ride.
They usually get the buddy pairs to jump in at about the same time so you stay together. The current at the surface moves faster than the current below, so you both need to be ready to descend quickly once you've jumped in and given your big ok. It will be calmer and slower below. Many sites in Coz have a slow, easy current. Some advanced sites have a much faster current.
The current is slower closer to the reef or structures. In the wide open, the current moves faster. You want to be pretty close to the reef or wall, but not extremely close to a wall, since currents can also move down a wall.
Usually in Coz, the boat follows everyone's bubbles and the Dive Guide has a marker. They tend to do multi-level dives, so everyone can off-gas pretty well. As people are getting lower on air and signal the DM, they ascend in buddy pairs and the boat picks them up wherever they surface. People keep going up two by two wherever they may be until the last couple of divers go up with the DM. Even during a night dive, the boats do a fabulous job of keeping track of the divers.
Safety gear is always a good precaution - a whistle, SCUBAlert, and Surface Marker Buoy at least. An electric or chemical light stick for night dives plus a dive light or two.
Before the first time I went to Coz, I asked questions similar to yours. I also found that reading the Drift Diving chapter in the AOW manual really helped as well.
Hope that helps. Have fun!