UPDATE: Cristian commented adding a nice solution (which I didn’t personally test though, so it’s on you):

1.  run chmod 775 on the download folder, with -R option (recursive on subfolders):
   sudo chmod -R ug+rw folderName
2.  add your own user (the one you need to be able to access the downloaded files) to debian-transmission group, or any other group that the transmission daemon belongs to:
   sudo usermod -a -G groupName userName

And that would be all.

Though, after some tinkering that occurred as of june 2019, I found that my original, following, solution has a -probably- nicer “feature” for someone: the config folder of transmission is saved in your own home folder, instead of /var/lib/transmission-daemon/blahblah, in my case this saved some waste of time.

This is the original article with the original solution.

I have a raspi3 running transmission daemon downloading to an external USB drive shared via Samba. I don’t want to keep using debian-transmission user with the daemon since just switching to my user account having the access rights to the external USB is much simpler.

I had it already nice and running before, but updating the daemon with apt-get messed everything (and lost all the running torrents as well) so I had to rediscover the procedure once again, and for posterity (and myself for future occasions) I’m writing it down here.

By the way all the missing torrents appeared again after I solved the issue.

So, here are the things you need to do:

1. run sudo service transmission-daemon stop just in case
2. edit /etc/init.d/transmission-daemon to have USER=username
3. edit /lib/systemd/system/transmission-daemon.service to the same effect
4. run sudo chown -R user:user /etc/transmission-daemon/
5. run sudo chown -R user:user /var/lib/transmission-daemon/

Done.

According to a comment of this article, this might not work for you, so instead you might want to follow this guide instead:

https://askubuntu.com/questions/261252/how-do-i-change-the-user-transmission-runs-under/544185#544185