If your external drives are formatted as ext4 (or another Linux-native filesystem), then they abide by UNIX permissions.
Therefore, on a single-user installation, it is better to mount them to directories under your user’s home directory, because then the mountpoint will be owned by your user, as opposed to if you mount them under /run/media/, which is a temporary filesystem that ceases to exist once the machine shuts down or reboots. And if you want to mount them to directories in the root directory, then you ought to know that only the root account has write access there.
Please read through the following two elaborate tutorials, and you will understand… 
- [Tutorial] Understanding and working with UNIX filesystems and permissions
- [Tutorial] Working with additional 'drives'
