sudo chown -R g+w /path/to/site1
I've been having this problem for the last day since I setup my droplet.
Similar to the OP, I have nginx running Ubuntu.
I have a user with sudo power "jim".
This user owns all the files and folders within the html directory that compromise the WordPress installation.
i've added this user to the www-data group, as well as the reverse so that.
jim@noise:/var/www/$ id jim uid=1000(jim) gid=1001(jim) groups=1001(jim),33(www-data) jim@noise:/var/www/$ id www-data uid=33(www-data) gid=33(www-data) groups=33(www-data),1001(jim)
Still when trying to upload media or plugins from my computer I will get the error that The uploaded file could not be moved to wp-content/uploads.
The only way I can get uploads to work is by changing the ownership of the files and folders to jim:www-data, but as the OP noted this creates a permissions issue when trying to do anything via FTP.
From reading various threads this seems to be a recurring issue.