do I need to use rooting powder? ....how far apart I should space the clippings?
Some people do use rooting powder, and perhaps it does assist rooting, though it's not essential.
The spacing should be around 9 inches to a foot (20cm to 30cm). But you could put in cuttings twice as close as that (4 inches apart), so that if some cuttings don't take, you'll still get a decent hedge.
... I do think privet looks best when it is dense....
That's true.
You could plant two rows, 6 inches apart and with the plants staggered, so the pattern looks like a zig-zag. That would produce a thicker and denser hedge, but that would require more land area. Probably not what you'd want to do on an allotment, although you could if planting around a house.