In 2v2, Rocket League strikes the perfect balance between offense and defense. There aren't enough people on the field to keep someone back as a designated defender, so everyone has to change roles on the fly. That calls upon fast and consistent teamwork between you and your partner.
Only send one person to the face-off. If you and your teammate charge the ball and either miss or get a bad break on the initial collision, nobody will be there to guard the goal. That gives the opposing team an easy breakaway.
So, one person should charge. What should the other person do? That depends on how you want to approach the face-off.
There's a high risk/high reward approach that you can try. Face-offs often kill the ball after a collision. If the two cars on both teams strike the ball at the same time (which happens more often than you might think) the ball doesn't go anywhere. That creates the opportunity for a quick follow-up shot on goal. If you're the car that hangs back on the face-off, follow the initial face-off striker closely. Then you'll be there to hit the dead ball toward the opposing goal.
It's important to get there first. The player on the other team may try the same thing. Whoever gets there first gets an easy shot on an open goal. Keep in mind that it can backfire completely, since nobody is watching your goal.
There's a less aggressive approach to 2v2 face-offs. Still only send one player for the face-off. The other player can grab a full boost in one of the corners to prepare for the impending play.
Don't bunch up
Don't play Rocket League like kids play soccer when they're first learning. It does your team no good to have both of your team's cars in one place. There are multiple benefits from spreading out. One of those benefits is the assist and goal.
Hammer the ball into the corner
The most common offensive play in Rocket League is to have one player hammer the ball into the corner so that it bounces or rides the wall around the bend toward the goal. When one player is teeing up the assist, the other should be in the center ready to take the shot. Don't follow behind the player headed toward the corner. Then nobody is in position to take a shot.
Defensive rotations are simple in 2v2. The person in goal will have obvious opportunities to clear the ball or save a shot. When that happens, it puts the goalie out of position. The teammate should get in goal as soon as the clear or save is hit. Then the roles switch.
Don't fight over the ball or with your teammate
Trust is essential to 2v2 Rocket League. Trust your opponent to make a play. Don't try to follow your teammate because you think they're going to screw up a shot. Go to where the ball is going to go, not where it is and always be ready for a pass or rebound.
Don't bicker with your teammate, either. You both have two opponents in front of you. It doesn't make any sense to turn the game into three-on-one.
If you're playing with friends, you should know how much they can take and dish out. This isn't the case when playing with strangers. Ripping a teammate for a missed save or shot is usually the beginning of the end of that game. A stranger will have no problem quitting on you on the field or just quitting the game completely. Just don't do it, and try to improve as the game progresses.
The best way to play 2v2 is with someone you know. You'll understand each other's callouts and in-game slang, and it just makes for a smooth partnership.
Rocket League makes it easy to communicate with strangers thanks to quick chat, which is handled on the D-pad. Each direction opens four similar commands to convey to the field. Up is reserved for Team Chat that only your teammate can see. Season 2 added more quick chat options you can enable in the options menu. Check them out, and see which ones suit your play style. Need Boost is always a good one to have.
Communicating on offense
These kinds of plays require constant communication, arguably the most important aspect of 2v2 play. This is where it helps to play with a friend, but quick chat helps, too, if you're playing with a stranger. It only takes one play where you both go for the ball because you didn't communicate for the other team to get an easy shot on an open goal.
Communicating on defense
When on defense, always have one person in goal and only one person in goal. The player not in goal should be making a play on the ball to try to steal it away from the opponent. When the ball heads toward the goal, communicate so both players aren't trying to save the same ball. This is easier said than done, since instinct kicks in when you're trying to make a last-second save.
Always say when you're headed to goal, in goal and not in goal. We can't stress how important this is. Say this so your teammate doesn't think the goal is empty, or doesn't try to fill the goal when you're already there. These simple callouts will strengthen your defense tremendously.