Difference between public and private sector?

Difference between public and private sector?
Ownership of the public sector of a certain country is shared equally by all citizens of that country. There should be certain laws that specify how does the PS get revenues and how does it spend them, how does it hire & fire employees, lowest and highest possible salaries ...etc.
While ownership of any private sector entity is limited to the persons who paid up the capital to form it. Those persons may or may not be citizens of that country. Each private sector entity could invent its own rules of hiring & firing and payouts ...etc.