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Baby FTP Server?
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➔ Baby FTP Server website
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2 reviews of this app found across Reddit:
One pair of tools I've found super useful: Baby FTP and Baby Web Server from Pablo Software. Standalone FTP and Web server, single exe style. Great for when you're having to update esoteric firmwares at client sites. I'm looking at you, Brocade SAN Switches!
I'd probably try something like this: http://www.pablosoftwaresolutions.com/html/baby_ftp_server.html
I haven't ever used that before, but you should be able to set that up on the "sending" computer to make it a server. Then you'd use a program like FileZilla to connect to that server and download the files. This way you get a direct connection between the two computers, and don't have to pass through a 3rd party service. It'll be as much bandwidth as the two computers can handle (minus losses during transmission).
If you have Linux experience, there's like a million options, but I don't think you would have asked in that case ;)