Certainly the 4-way stop is one of the biggest challenges presented in group riding. The answer of what to do in this situation is: it depends on the situation, and also how competent the group is.

First, at any stop sign it is best for the leader to wait at the sign until the whole group is stopped behind him, even if it means he has to wave cross traffic through.
Second, as at any stop sign, typically the riders would advance two at a time. However, if the group is sharp at competent riding in tight formation, they could advance four at a time.

All this is fairly simple, unless stopped cross traffic is turning into the same direction we are traveling. Then you can get a situation where there is a car separating the group. Here the leader must make a decision. He can:
1. Wave traffic through, and proceed behind. This works if there is just one car.
2. Signal the car driver to let us go first. If you can get their attention, most car drivers offer this courtesy to a large group. Since riders in the rear won't know of this agreement, the second rider can remain at the sign and wave the others through, moving back up to his slot on the right, as in a "sweep" maneuver. This takes some experience riding together.
3. Proceed in small groups, assuming that the group(s) in the rear can pass and catch up. This is a pretty good bet on open country roads with little traffic.
4. Proceed in small groups, and re-convene in a parking lot or along a wide shoulder. This sometimes happens on busier roads.

As you can see, there are many variables, and we never know just how any situation will turn out. So we fall back on the basic rules:
a. You are responsible for your own safety. Be aware of what is going on, and if a situation doesn't feel safe to you, do what you believe is the safest thing.
b. Pay attention to what the group is doing, and go with the flow, bearing in mind that item "a." above always takes precedence.
c. If the group gets separated, who ever is at the lead of the second (and third and fourth) groups becomes leader of that group. If that's you, think like a leader! If you are uncomfortable, signal a more experienced rider to take the lead for you.
d. Don't worry! We haven't lost anyone yet (though we have had a few folks bewildered for an extended period.) Sooner or later we will all get back together again.

Sincerely, Mr. Answer-Man