I did some research and asked the guys I co-op with. I determined the following:
- The person hosting the game's progress determines which story mission comes next. If you finish a story mission in co-op that is ahead of your own story process, when you catch up (in single player or if you're the host next time) you'll have the option to skip this mission. If the mission unlocks something (ie, a new crib) you'll have access to this in your game, regardless of whether or not you've gotten there yourself.
- Activities and other side missions count for both people if finished in co-op.
- Challenges in the Saintsbook and controlling parts of the city are determined separately, co-op or not. For example, if you want the bonus for owning a store, you have to buy that store.
- Assassinations count for both people, if you both have any target selected. It doesn't seem to matter if you've got the same one selected, so long as you both have one marked. I got credit for 2 or 3 that my buddy was working on halfway across the map, simply because I was working on a different target.
- Vehicle deliveries can only be active on one person at a time. If you both try to select a vehicle to steal that your partner already has marked, the game will tell you that it's already in progress.
- Collectibles (ie, cash/drugs/photo ops/etc) are tracked separately.
- Disrupting gang operations and completing "survival" challenges counts for both people, if both people were inside of the radius of the operation/challenge when it finished.
You have separate garages, and a vehicle that came from your partner's garage can't be stored in your garage. (ie, if you steal a tank and put it in your garage, you can't then get the tank out and let him drive it to his garage to get a copy)
I finished the story missions in a co-op game, and I don't have the option to replay the final mission over again. I don't know if this is a local glitch or a problem with finishing the story in co-op, but I thought I'd add that as a note.