The council really should be more specific when they use the word "private".
In this case they are saying that this estate is not a Local Authority estate.
Councils cut the grass and maintain the open spaces on Local Authority estates.
Councils generally do not cut the grass and maintain the open spaces on non Local Authority (what they call private ) estates.
When a council takes an estate in charge they generally take the roads, footpaths, public lighting, storm water, mains water.
The responsibility for the upkeep of the green areas and open spaces rests with the developer / management company / owners.