Social Butterfly

You took part in an online playlist and stuck around for at least 5 races.
Trophy Type: Bronze

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Straight-forward. All you need to do is get in a online playlist and play at least 5 races. After the 5th, the trophy will pop.

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This video also shows the trophy "All Night Long" being unlocked so here is the info for that trophy as well.

All Night Long

You took part in and completed an online Endurance race.
Trophy Type: Gold

Self-explanatory. Go to Multiplayer and search for an Endurance race, complete it and the trophy should pop when you get back to the menu. You dont have to win, just complete it without crashing your car.