Private object has unknown nature (11)
I believe the answer is:
personality
'nature' is the definition.
(thesaurus)
'private object has unknown' is the wordplay.
'private' becomes 'personal' ('personal' can be similar in meaning to 'private'**).
'object' becomes 'it' (pronoun for an inanimate object).
'has' means one lot of letters go next to another.
'unknown' becomes 'y' (algebra).
'personal'+'it'+'y'='PERSONALITY'
(Other definitions for personality that I've seen before include "Individual character" , "(Distinctness of) character" , "Charisma" , "Distinctive character" , "Celebrity" .)