You have probably found out by now, but just in case...
by the looks of it you can only any of the side missions. It's is barred from the main levels due to the co-op character being an npc in some of the missions.
(there seems to be 16 side missions, 2 of them pre-order bonuses, so future dlc :'( )
Whenever you start a mission you have the option of doing co-op, which lets you either invite a mate, or wait for a random to join your game or you his.
You cannot join an already in progress mission (at least i cant see how to join an in progress one)
Mission wise having the second person allows you to progress sometimes on a SLIGHTLY different path due to a context sensitive action, eg: dual boost/breach need to stress, in my experience the dual actions set you up for a slightly different route. for example.
Mission A, if i had a partner i could've dual-boosted up to an area allowing me to kill a very annoying sniper.
as i didn't i had to work my way around, picking my times to ghost the people along the way as i didn't want to risk shooting and missing him alerting others.
Mission B, a dual breach, after exploring the level thoroughly it seems a dual breach allows potential me & co to enter a room from the right instead of the left, making it easier to sneak by 2 people.
Also after you have acquired your flying fortress of doom one of your team-mates offers co-op only missions (4 of total 14/16 optional missions).
Gameplay-wise other then the context sensitive actions, you have the dual execute feature.
Some evil enemies wear helmets (along with other annoying things). The dual execute allows one of you to shoot off the helmet, and the other shoots him in the head. With the normal execute you just shoot off the helmet.
Hope it helps