Your friends rig is more powerful than your rig, although if you get a better cooling solution than your friend, you could overclock your CPU, GPU and RAM, you might be on par with his rig then.
You're going to have a hard time matching his performance because the 580 is substantially more powerful than the 6970 (and priced accordingly). You do have a fairly high end rig though- you NEED to OC your processor. If you have the budget you could also consider getting another 6970 and 4GB additional RAM.
------------------------------Core i7-950 @ 3.91 GHz | Gigabyte UD3R | 6GB Kingston HyperX DDR3-1600 | GTX 560 ti SLI | Corsair F120 SSD | OCZ Agility 3 240GB SSD | Samsung Spinpoint F3 1TB HDD | Zalman CNPS10X Quiet CPU Cooler | Corsair 750TX PSU | Coolermaster HAF X Case | HP ZR30W Reply to beltzy
BF3 is not a cpu intensive game so I wouldn't worry too much about overclocking the i5 (although you should anyway)
I would give oc'ing the 6970 a shot and for a graphically intense game like BF3 you may seriously want to consider a xfire set up. In addition I would go up to 8 gig ram which should help slightly.
So i tried my friends 580 the other day and noticed the difference. When i put back my 6970 in i saw it had a switch on top. I researched and found out it had to do with flashing the bios. I ended up switching it to the other side to see what would happen. Wouldnt you know it, it played about as good as the 580.