Fast, easy, reliable way to transfer large files between Android devices? (aka. WiFi Direct stories from the field)
Rather than sleeping, I spent a few more hours experimenting.

I found what works best for my needs is an FTP server app. 

For casual transfers, when both devices are connected to the same network, I'll just transfer through my Access Point. With the limitations that implies.

If I'm not near an AP or just want full speed, I'll deal with the hassle of setting of the Wi-Fi direct partnership and then launch the FTP server. It usually takes 3-5 tries before the devices connect, but once they do, FTP works stable and fast.

Furthermore, this solution also satisfies my secondary goal, of being able to use the same method to transfer files between Android device and PC.

Now, there are a lot of FTP server apps out there. Most are ad-supported, most are rather big as well. To my surprise, there's also quite a performance difference between them.

I thought I'd might as well shared my experience with the four FTP server apps I tested...

Required Permissions

None of the tested apps required ridiculous permissions. Just the following special permissions:


FULL NETWORK ACCESS
MODIFY OR DELETE THE CONTENTS OF YOUR USB STORAGE MODIFY OR DELETE THE CONTENTS OF YOUR SD CARD
PREVENT TABLET FROM SLEEPING PREVENT PHONE FROM SLEEPING  


In addition, Andreas Liebig's FTPServer also required:


READ PHONE STATUS AND IDENTITY    


Speed Test Setup

The speed test wasn't exactly scientifically performed, but devices were in the exact same spots during tests.


FTP server app running on Nexus 7
FileZilla FTP client running on Windows 8 laptop
Both connected to the same 802.11n network. Windows reported the link speed as 130 Mbps (I know this is not exact)
I downloaded the same 103,4 MiB large file two times through each app


Test Results

All stats, numbers and other facts are as of March 30th 2013.

Apps ordered by measured throughput, fastest on top.

Andreas Liebig's FTPServer


Price: Free
Latest version: 2.4.0 (2013-03-26)
Size: 78 kB
Install base: 100.000 - 500.000
Avg. rating: 4.6
Android OS requirement: 1.1+
Comments: Initial configuration required, not plug-and-play. Detailed server log on main screen when service is running. No anonymous access possible?    
Throughput: 103.4 MiB downloaded in 43 seconds @2.4 MiB/sec, 103.4 MiB downloaded in 47 seconds @2.2 MiB/sec


Boby Ertanto's My FTP Server


Price: Free (ad supported, ad-free version available)
Latest version: 2.1 (2013-02-28)
Size: 1.2 MB
Install base: 50.000-100.000
Avg. rating: 4.1
Android OS requirement: 2.2+
Comments: Simple interface. Very basic settings.
Throughput: 103.4 MiB downloaded in 48 seconds @2.2 MiB/sec, 103.4 MiB downloaded in 49 seconds @2.1 MiB/sec


The Olive Tree's Ftp Server


Price: Free (ad supported, ad-free version available)
Latest version: 1.11 (2013-01-14)
Size: 1.1 MB
Install base: 100.000 - 500.000
Avg. rating: 4.4
Android OS requirement: 2.1+
Comments: Based on Apache ftp server. Simple interface. Configurable port, home dir, username/password, service resillience, energy saving etc.
Throughput: 103.4 MiB downloaded in 56 seconds @1.8 MiB/sec, 103.4 MiB downloaded in 56 seconds @1.8 MiB/sec


Berserker's FTPDroid


Price: Free (ad supported, ad-free version available)
Latest version: 1.3.1 (updated 2012-12-28)
Size: 2.0 MB
Install base: 50.000 - 100.000
Avg. rating: 3.9
Android OS requirement: 2.2+
Comments: Based on  Pure-FTPd. Simple interface. Configurable port, home dir, username/password, service resillience, energy saving etc.  
Troughput: 103,4 MiB downloaded in 135 seconds @0,8 MiB/sec, 103,4 MiB downloaded in 121 seconds @0,9 MiB/sec


Conclusion

Not only is Andreas Liebig's FTPServer the only completely free of the apps tested, it's incredibly light weight (78 kB!), can run on any Android phone you can find but also has the fastest throughput of the apps tested.

Once the initial configuration is done, the main window simply displays your standard FTP server log, whereas other applications pretty much displays nothing but a few buttons and maybe the IP address and port of the server.

Thank you, Andreas!