Test footage details:
The choice of lenses was done very carefully. Considering different crops of the APS-C and micro-four-thirds sensors (1,5x vs 1,86x recpectivelly) I've used the lenses that have similar view angle, same speed and extremely similar optical performance. All lenses had proper lens hood at all times. The aperture was most of the time stopped down for one step (day at f/4.0, night at f/2.8) except in very dark scenes, where both Nikkors had fully open diaphragm. In the NEX-7 there is no possibility of shooting in 24fps mode so I set it to 25fps and adjusted the GH2 to HBR and 25fps.

NEX-7 with stock firmware (PAL) // GH2 with the patch Cake 2.3 by @balazer
AVCHD 25fps 1080 // AVCHD HBR 25fps 1080
The GH2 camera files were quite exactly double-sized compared to the NEX-7

Daylight footage settings:
Lens: Canon FD 35/2.8 // FD 24/2.8
Lens aperture: (always) 4.0, no filters
Crop: 1.5x35=52,5mm equivalent // 1.86x24=44,6mm equivalent
ISO: 200 // 200 (always)
AWB // AWB
EV 0.0 // 0.0
Metering: multi // frame+center
Colors progr.: standard // standard
In-camera setting: default // default (0,0,0,0)
Mode: aperture priority (both)

Night footage settings:
Lens: Nikkor 35/2.0 Ai // Nikkor 28/2.0 Ai
Crop: 1.5x35=52,5mm equivalent // 1.86x28=52,1mm equivalent
Lens aperture: 2.0-2.8
ISO: 1600-1600
AWB // AWB (one take manually set to 2500K both)
EV: at first 0.0 both, later -0.3 / +0.7
Metering: multi // field+center
Colors progr.: standard // standard
In-camera setting: default-default (0,0,0,0)
Mode: manual (both)

Comments and bit of my subjective impressions:

Exposure: by default settings (EV 0.0), NEX-7 tends to overexpose EV+2/3, GH2 to underexpose EV-2/3
Colors: NEX-7 tends to reddish, GH2 to greenish colors. The NEX-7 has possibility of adjusting the color tones for every color mode, also for AWB.
White balance: NEX-7 is capable of getting much cooler AWB, extremely weird situation occurs at 2m45" when the WB in both cameras was set manually to 2500K
Moiré: two tipical situations by 1m40" but generally, the NEX-7 has obviously less moiré
ISO: NEX-7 has feature of changing the ISO sensitivity during the recording, the GH3 offers this but GH2 and GH1 not. In the daylight ISO was set manually to 200, in the night to 1600 in both cameras.
Sharpness: I was expecting clear winner GH2 and much worse NEX-7. Yet, the result is very similar. Please mind that in one scene I forgot to focus the lens (by 2m35") mounted to the NEX-7
Contrasts: in all test situations the NEX-7 shows more details in the shadows, but it would need bit more testing because of the camera's propensity to overexpose what I didn't notice at first.

The NEX-7 is even smaller than GH1 or GH2 and it has bigger sensor.... I like its concept, very minimalistic- pure idea of the "mirrorlessity". Its big advantage is the still photography, such a comparison wouldn't make much sence because it simply blows off not only the GH2 but even most of the DSLRs with APS-C sensor.... it's newer generation. The big drawback of the NEX-7 considering its video functionality is its sensor overheating. On the room temperature, some 20min are maximum one can expect, regardless if the display is lifted or left in its position. I am kind of very pleased to see that the NEX-7 renders the shadows very smooth and in cinematic spirit, with lots of details. It is a feature that we are struggling to get with our GHx cameras. This cinematic image gets more obvious by using a bit less sharp lenses than Canon FD or Nikkor that I used in this test.