I was seeing this yesterday.
You need to check the ownership on the drive. A lot depends on the actual folder that OS X uses for the Trash, and what permissions it has.
To get it to work for me, I need to:
1) Ensure "ignore ownership on this volume" was NOT checked
2) That my username (i.e. login name) was added as someone who could read and write to the volume (this wasn't the default).
Once I did that, trash from Lightroom worked. Note this isn't a Lightroom problem per se, LR is just calling an API to send something to the trash, and if permissions aren't right, that won't work and LR punts and just deletes things, rather than trying to fix them (which may not be possible to do).