I am experiencing a very similar problem.. My gy-521 worked very well a few days ago with an arduino uno, but with a nano it stopped working and i couldn't get it working with the uno again. Green LED still lights up, but it doesnt show up in any i2c scanners and it does show life if i connect ONLY i2c pins to 1 arduino and i connect another arduino properly. I have posted it here too: MPU6050 acting very weird.... - Sensors - Arduino Forum ..
i am playing with it for a few days now, but i haven't got it to work still. It's just a guess, but i'm guessing it might need to get some kind of a different i2c command to send a it's data.
You can try to put a wire between INT (interrupt) and D2. When I've tested, it didn't require Int pint to be connected to D2, but after whole day of trying to fix it, it only showed life, when i connected it, but to me it's still not working properly.. it sometimes work, but mostly doesn't....