There is a lot of confusion surrounding the 'paupers grave' terminology.
They are most likely buried in a 'common grave' with several unrelated people.
Although the grave is not owned by the family, their funerals would have been paid for (coffin, service, flowers), just not including the full price of the burial plot.
Our ancestors would be horrified to think of being buried in a true 'paupers grave' (one where today the state pays for everything - usually a completely no-frills cremation), they would have scrimped and saved to get the money together for a decent funeral but the cost of the plot was and still can be out of the financial reach of many families even today.
Dawn