I've been in several block swaps with a small guild I'm in. We get together and decide what we're going to do and what the rules are for the particular swap. There are round robins where each person adds a "round" to the center. There are row robins where each person adds a row of blocks. There are block exchanges where you make a set number of blocks for each person.
Each one of the swaps can have any number of rules too. Like on the round robin you can make a rule for each round such as round 1 has to be appliqued, round 2 is to be pieced, round 3 has to have triangles, etc, etc. On the row robin you can do the same sort of thing but make it like row 1 is to be flower blocks, row 2 is to be house blocks, etc, etc.
For block swaps we did one were where you put your main fabrics in a plastic tub and then pass to the next person and they can make whatever block they want but they have to use at least two of your fabrics and can add one of their own if they wish. We also did a log cabin block swap where each person added three logs from their own fabric stash to the blocks. For that one each person started their blocks with the center and two logs added and then put in instructions for what they wanted added. One person requested patriotic, one person requested girl colors on the light side and boy colors on the dark side. Everybody ended up with some really pretty log cabin quilts out of that swap.