Connecting my devices with Mac as the "server"
If you share files from your Mac by checking the box next to 'File Sharing' in the 'Sharing' pane of 'System Preferences' then you need to connect to that share using a client app on your device (like the 'File Connect' app for iOS.) Apple File Protocol (AFP) and Microsoft compatible SMB are the two choices for sharing.

Bittorent is a peer-to-peer method of sharing files. AFP, SMB, and FTP are client/server methods of sharing files and work completely different from peer-to-peer. With a client/server method, you would access your files directly from your Mac.

For what it’s worth, if DropBox is not working out for you, the problem is likely on your end.