A file can only be owned by one user and grouped to one group. chown changes the user but can't add users. Optionally you can use chown to set the group for the file by adding a colon after the user and putting the group there. That's not what you want to do.
Putting both users in the same group is the approach you need, combined with suitable permissions. Ideally, this shouldn't be in a user's /home/user/ directory since that is supposed to be private to the user (700 permission) and other users shouldn't be able to access anything beneath it. You should really create a shared directory (say /home/shared/) for this and set its permissions to 770 to allow its owner and group members to write to it. This shared directory can be owned by one of the users but the group depends on how you want to share access.
The next thing is to decide whether you want the second user to have access only to the files in the shared directory or potentially anything grouped to the first user. If the latter you can simply add the first user to the second user's primary group. Each user will need a umask of 002, 006 or 007 to ensure files are stored by default with permissions that will allow full group access. Directories need to have 774, 771 or 770 permissions and files 664 or 660 permissions which these umasks will achieve.
If you want more limited access you can create a special group for the sharing (say "sharegrp") and add both users to it. In that case you would need to change the group ownership of each file on creation to sharegrp. However, that's probably more cumbersome than you need.
You probably only want to do this one way round since both users will have full access if they are members of the group and either not the owner or the owner has access also. However, if both users will create files you might want to do a reciprocal membership to ensure the file is accessible to both whichever created it.
I'm not sure it matters which interface you use, but the user is always added to the group, so if you add user DAN to group JON, DAN will be able to access anything the JON group has permission to access unless he is the owner and the owner is forbidden (which is very unlikely).
So if file x has owner:group of JON:JON and permissions of 660 (-rw-rw----) DAN and JON could both read and modify it (JON as owner and DAN as group member). If it had 640 permission both could read it but only JON could modify it and if it had 460 permission both could read but only DAN could modify it (but it is very rare to give the owner less permission than the group). On the other hand, if the file were owned:grouped by someuser:JON, both JON and DAN would access it according to the group permissions of the file.
Hope that's clear.