Grant a user permissions on www-data owned /var/www

If we check ownership of site1, we will find something like this,

ls -ld /var/www/site1/
drwxr-xr-x 2 root root 4096 Oct 24 21:06 site1/

This means that the directory is owned by user root, group root. While user root has write permission (plus read and execute permissions) to the directory, group root has only read and execute permissions.

We will want to change the group ownership to another (new) group and add user1 to that particular group. We will give write permission to that particular group as well.

Create a new group,

sudo addgroup site1

Add user1 to the newly created group,

sudo adduser user1 site1

Check that user1 is really in that group,

groups user1

The output should be a list something like,

user1 : <other-groups> site1

Now we can change the group ownership of your intended directory.

sudo chown -vR :site1 /var/www/site1/
changed ownership of `/var/www/site1/' from root:root to :site1

Grant write permission to this new group owner,

sudo chmod -vR g+w /var/www/site1/
mode of `/var/www/site1/' changed from 0755 (rwxr-xr-x) to 0775 (rwxrwxr-x)

Check that all the changes are indeed there,

ls -ld /var/www/site1/
drwxrwxr-x 2 root site1 4096 Oct 24 21:06 /var/www/site1/

So, the directory now is owned by user root, group site1. Both user root and group site1 have write permission (plus read and execute permissions) to the directory. Any user belonging to group site1 will enjoy all the privileges granted to that group.

Now login as user1, move to site1 directory and try to create a file in that directory,

echo "My User1 Site" > index.html 
bash: index.html: Permission denied

This failed since most likely the primary group of user1 is not site1. So, change to that group.

newgrp - site1

Try to recreate the file (beware that you have been moved to the home directory of user1 after changing group), this should work now. Since the created files will have world read permission, apache (or your web server) should not face any problem accessing them.

EDIT

Also, as pointed out by dan08 in comment, you need to add www-data to site1 group.

sudo adduser www-data site1

On many (not all) distributions, www-data is the user under which the Apache web server runs. This also means that everything done by Apache (especially including PHP scripts) will be done with the permissions of user www-data (and also group www-data) by default. WordPress uses the user www-data to write files.

If you want to see how apache web server is running, issue the command,

ps aux | grep apache2 | less

For those who have their wordpress root folder under their home folder:

Ubuntu/apache

1. Add your user to www-data group:

   CREDIT Granting write permissions to www-data group

   You want to call usermod on your user. So that would be:

   sudo usermod -aG www-data yourUserName
   

   Assuming www-data group exists

2. Check your user is in www-data group:

   groups yourUserName
   

   You should get something like:

   yourUserName : yourUserGroupName www-data
   

   yourUserGroupName is usually similar to you user name

3. Recursively change group ownership of the folder keeping your user ownership

   chown yourUserName:www-data -R yourWebSiteFolder/*
   

4. Change directory to yourWebSiteFolder

   cd yourWebSiteFolder
   

5. Recursively change group premissions of the folders and sub-folders to enable write permissions:

   find . -type d -exec chmod -R 775 {} \;
   

   mode of /home/yourUserName/yourWebSiteFolder/' changed from 0755 (rwxr-xr-x) to 0775 (rwxrwxr-x)

6. Recursively change group premissions of the files and sub-files to enable write permissions:

   find . -type f -exec chmod -R 664 {} \;
   

   The result should look something like:

   WAS:
   -rw-r--r--  1 yourUserName www-data  7192 Oct  4 00:03 filename.html
   CHANGED TO:
   -rw-rw-r--  1 yourUserName www-data  7192 Oct  4 00:03 filename.html
   

   Equivalent to:

   chmod -R ug+rw foldername
   

   Permissions will be like 664 or 775.

Create two groups: site1grp and site2grp

sudo groupadd site1grp && sudo groupadd site2grp

Add www-data to both groups.

sudo adduser www-data site1grp && sudo adduser www-data site2grp

Add user1 and user2 to the appropriate groups

sudo adduser user1 site1grp && sudo adduser user2 site2grp

Change the permission of your site folders so that the user owner is www-data and the group owner is the appropriate group

sudo chown -R www-data:site1grp /var/www/site1  && sudo chown -R www-data:site2grp /var/www/site2

Now www-data has the user and group permissions on both sites and each user has the group permissions for their respective site.