You can start up a match in a lobby, ala left 4 dead, with friends and then go from there. To get a guaranteed cosmetic drop, exclusive to mann v. machine mode, after completing a full game you need to have a mann up ticket to get on a mann up server, which are $.99 each. Those are, I believe, the games played on official valve servers. The training mode servers can be any but while you won't get the cosmetic items, normal weapons still drop. Also, don't know if this is true or not, but the game I played on the mann up server was significantly harder than the training server, and in the mann up game we were much more coordinated and still struggled through it.
Meant to mention this in my last post but the new canteans are a god send late game, specifically the insta-upgrade one that the engineer can get.