How to thicken up a privet hedge...

We have a privet hedge at the front of our house.  Bizarrely on one side of the entrance it loses all of its leaves in winter and the other side, it is evergreen.

The side that loses its leaves is under a huge beech tree, so I add fish blood and bone, and mulch with manure once or twice a year. 

I have been advised to cut it back by a third in order to promote new growth as when it does get its leaves its not very thick and from a privacy point of view not great.

When would be the best time to thin it out to get new growth.  I'm thinking I should do it around now.  Do i cut it back, then mulch and water.  Will i achieve a thick hedge under such a large tree ?

Any advice gratefully received.  thank you.