How do I use gnome trash for files in different partition?

When you delete a file on a removable storage device, GNOME automatically creates a trash folder at the device's root. If your user ID is 1000, it will create a file named /data/.Trash-1000. When you go to the Trash place in Nautilus, you see an amalgamation of all the trash folders for your user on all the filesystems that are connected.

It sounds like this one is a permissions issue. GNOME is trying to create the trash folder at the device's root, but perhaps your user only has access to individual files under the root. The easiest way about this is to make the thing more accessible. In a terminal, try sudo chmod 777 /data to make the root of your data drive editable to any user. Alternatively, you can do sudo chown yourname /data for a similar effect.

GNOMEish file managers need a place to put the trashed files. Copying them to the 1st partition would be very bad performance. So it tries to put them in the /.Trash-$UID folder. Without rw access to that folder, no trash.

Run this bash in the partition root as the user who needs a trash.

sudo mkdir .Trash-$UID && sudo chown $USER:$USER .Trash-$UID

Note: If there is a a rw .Trash folder, Nautilus may create the user's trash folder automatically (/.Trash/$UID). I haven't tried this, just saw it in some logs. There are security issues with public trashes. YMMV.