How do I determine how full a remote volume is? I use a remote volume on my wife's Mac Mini as a shared music library. How do I know how full that volume is getting without logging into her machine? Get Info is unavailable.
To get those informations mount the shared volume(s), open Terminal.app, enter df and hit Enter.

You will get a result similar to that one:

Filesystem                              512-blocks       Used  Available Capacity  Mounted on
/dev/disk0s2                            1463469952  960523624  502434328    66%    /
devfs                                          743        743          0   100%    /dev
map -hosts                                       0          0          0   100%    /net
map auto_home                                    0          0          0   100%    /home
map -fstab                                       0          0          0   100%    /Network/Servers
/dev/disk1s2                            7813298080 7407284144  406013936    95%    /Volumes/TimeMachine
afp_5Z456sdchuia84afaufhjdsg-1.2d000004 7813093280 7196533600  616559680    93%    /Volumes/SharedVolume


If you enter df -h instead of df you will get "human-readable" sizes. Those numbers are in TiB/GiB/MiB instead of TB/GB/MB though. Finder shows the sizes in TB/GB/MB.

In the example /dev/disk0s2 is the internal system volume mounted to /,
/dev/disk1s2 is an external USB drive with a volume mounted to /TimeMachine
and afp_5Z456sdchuia84afaufhjdsg-1.2d000004 is an AFP-share mounted to SharedVolume. Your names may vary, especially the filesystem name (afp_....) of the shared volume on the remote Mac.

The volume in question is the last one starting with afp_ which is the shared volume mounted to your desktop containing the shared music library. Use the name of the mount point to determine the relevant share if you have several entries.

All numbers (except the "Capacity") are blocks à 512 Byte.
So in my example the shared volume ("SharedVolume") has a size of 4000,3 GB with 3684,6 GB occupied and 315,7 GB free. The used capacity is ~93%.