How can I encourage my PRIVET to shoot new growth?

Not sure what you mean - its too early for it to be growing now anyway, it's still winter, so unless you mean you want to thicken it up during the season, which might be achieved by cutting, then I don't know what to say.

You can cut privet back very hard, but it will not leaf up well in a very shaded area. I have an hedge facing east and shaded on the other side by a row of sycamores. The east facing side is always nice and green but the growth on the shaded side is very sparse. You would stand more of a chance if you reduced the shade cast by the bamboo. Privet and bamboo are both greedy feeders too, so they may well be in a bit of competition

Ah, I see you've added more to your original question. Agree with Steragram's answer above.