After I chopped it down, I planted daffodils (Falconet and Sweetness) under the remains of my privet. It gets the morning sun only but the daffs have been great this year (in spite of spending last winter under builders' rubble).
Cyclamen are supposed to do well under privet (they don't survive under builders' rubble)

If you choose 'woodland' plants, the amount of light won't matter so much, but dig in some organic material help things along by regular feeding and watering.

Privit is a gross feeder so as Urbanite has already stated you will need to dig in plenty of organic material and feed and water regularly.