Privet is a greedy feeder and will take all the nutriments and the water from the soil so attempting to grow anything near it is going to be a challenge. Take a look at the soil near your privet hedge it is likely to be rather dry and 'poor' looking.

I planted hosta, keeping them well watered they did well. A few weeks ago I moved all the hosta to another part of the garden, reason being the hedges were being removed.

Bear in mind that you will probably need to cut the privet on a regular basis so anything nearby will have to withstand walking on (eg grass) or otherwise be clear from potential damage.
I put cyclamen hederifolium and spring bulbs in my front border when I chopped down the privet hedge. Hedge survived in spite of my effort and is now about two foot high and still horrible - nothing will ever convince me that privet should have a place in the (sub)urban garden landscape.

I'd go with cyclamen hederifolium as Urbanite said, and you may also find that Brunnera will thrive, although you could then be digging it out for years to come.

Urbanite… Bulba cut our privet hedge (agree horrible thing!) to the ground and treated the stumps with root killer - worked a treat! The hedge was a good metre wide meaning we reclaimed lots of ground. Needed plenty of organic matter to start with but within a few years we had good soil to grow plants in...

If you want to plant close to the hedge and have nothing in front of a narrow bed then there's a very thuggish pink flowered perennial geranium which will survive anywhere and spread. I began with one eight years ago and now they cover a whole bank, taking over from rank grass..

If you want one or two PM me in the autumn...