Mods and cars in multiplayer are separate from the singleplayer. You'll first have to unlock Standard Mods by completing certain milestones in each car. After unlocking a Standard Mod, a new milestone for unlocking the Pro version will appear. Unlike the singleplayer Pro mod achievement, you only need to unlock 3 in multiplayer. Choose 3 that look easy and go for them.
The majority of them can be unlocked without much effort, just by playing in public games online. I was able to unlock all the Standard Mods after playing for a few hours without going out of my way.
Off-Road - Jump 1,500 yards with Reinforced Chassis equipped.
Track - Drive 25 miles.
Reinflates - Drive 50 miles.
Burn - Free
Powershot - Drive 100 miles.
Lightweight - Get 60 Near Misses with Short Gears equipped.
Reinforced - Drive 200 miles.
Aero - Drive 5 miles.
Impact Protection - Get 10 takedowns with Powershot equipped.
Long Gears - Drift 5,000 yards with Track Tires equipped.
Short Gears - Drive 10 miles
Off-Road Pro - Drive 60 miles Off-Road with Off-Road tires during Speedlists.
Track Pro - Drive 80 miles above 100MPH with Track Tires during SpeedLists.
Reinflates Pro - Reinflate 32 tires during SpeedLists
Burn Pro - Get Perfect Ntrious 40 times with Burn Nitrous during SpeedLists.
Powershot Pro - Get 50 Nitrous-powered Takedowns with Powershot equipped during SpeedLists.
Lightweight Pro - Jump 15,000 yards with Lightweight Chassis during Speedlists.
Reinforced Pro - Get 50 Takedowns with the Reinforced Chassis during Speedlists.
Aero Pro - Slipstream 20 miles with Aero Body during SpeedLists.
Impact Protection Pro - Get 20 Slam Takedowns with Impact Protection during SpeedLists.
Long Gears - Go through 40 Speed Cameras above 100MPH with Long Gears equipped during SpeedLists.
Short Gears - Record a 100 yard Drift 160 times with Short Gears equipped during SpeedLists.