Owncloud provides the "dropbox" clone without using btsync, but if you want to add secure transfer - I understand that may be a valid addition to the mix.
That being said, your permissions for the file system will need to be enabled for the user running the OwnCloud server and also btsync. Typically this will be
www-data on a Debian Apache server and
btsync for the btsync portion. It may differ if you are using other than default components such as Nginx etc.
Set the permissions and ownership using
chown, and make sure that the user running the btsync is also in the
www-data group so that they can both access the files. To add a user to a group in raspbian, you can use something like
adduser btsync www-data to accomplish this. As with almost all commands in Linux, you can use either/both of the options for syntax help -
man <command> or (usually)
<command> -h or
That should do the trick. But if all else fails, temporarily open permissions up to all users for the target files (
chmod -R ugo+rw <folder>) and then let the system run for a moment and check what/who it is actually writing files as - then lock it back down to that user/group, using the commands above.