Non-profit org needs file transfer software recommendation

I'm volunteering for a non-profit organization based in Chicago that collects community support for global initiatives through the collection of digital signatures and related supporter created profiles & data. We often have to transfer large files, more than several gigabytes, to organizations located throughout the world. The problem is that some of these organization are located in countries with a weaker IT infrastructure and large file transfers can take a very long time or the transfer may completely fail due to the unstable network. Mailing hard drives or using most free transfer products isn't possible because we need to ensure that we are securing our supporters data and privacy. Still, As we are a non-profit, we don't have extensive funds to commit towards software. Does anyone know of such a software product that could solve this issue? Any suggestion or advice would be greatly appreciated.