How is one supposed to make the distinction between when it is his and when it is hers?
well here it can be both but you can also say
pâinea nu este a lui = the bread is not his
pâinea nu este a ei = the bread is not hers
In British English (at least) "the bread is not theirs" CAN mean the same thing. Why we use the plural I'm not sure, but it often happens if gender uncertain or I sense when we're being more polite.