To remove a privet hedge from your property, you must also remove its roots. If left in the ground they will eventually decompose. But that may take decades. And you will not be able to plant anything in that spot in the meantime. If you have more immediate plans for that section of your garden, there's no getting around digging out those roots.
Cut back the privet hedge's foliage to within a few centimetres of the ground with your lopping shears.
Water the ground with a few centimetres of water. Moist soil is much easier to dig than dry soil.
Dig a trench around the privet hedge's roots to expose the edge of the root ball, using a spade. If you encounter so many thick roots that it is tough to dig, you're digging too close. Widen your circle. Dig until you reach the depth of the root ball.
Wedge your spade into the trench to undercut some of the roots underneath the root ball you just excavated. Push in as deep as you can. Use your foot on the top of the spade to give added pressure. Move around the entire circumference of the root ball.
Lift the root ball out of the hole. Use the blade of your spade as a lever. Wedge it underneath the root ball, then push the handle down to lift the root ball. Repeat on all sides until the root ball is pushed out of the hole.
Refill the hole with a few bags of commercial garden soil.
Invest in a quality spade to make removing the roots easier. Use a lopper to slice off the main tap roots underground.