Either would do but as its between the houses you might find privet is easier to keep narrower. It is sort of evergreen but does lose some of its leaves over the winter and can lose them all in very cold conditions. There is also Lonicera Nitida which has tiny leaves but grows very densely if clipped regularly. There is a plain green one and a golden yellow one called Baggeson's Gold, which would be lighter for you in that situation.

I'd make two points about lonicera nitida. It's very fast growing; to keep it tidy you'll need to trim it every couple of months. And it's not rally suitable for growing to six feet in my opinion.
Despite its reputation for being slow, yew makes a good hedge.
Another possibility is beech (or hornbeam on heavy soil). Although not evergreen, both hang on to their brown leaves all winter, making a screen you get the merest glimpses through.

I would not choose laurel, it grows fast and get big and strong and difficult to control, privet would be easier to look after.

I would go for beech or hornbeam as suggested above.
They make for very good screening and are a magnet for all manner of birds.
Not evergreen, but (ever)brown in winter and (ever)green in summer. I think just one annual trim for shape and bob's your uncle... :)

Hornbeam is also good if you need to keep it thin - I have seen Hornbeam hedges only 8 inches thick.

Point taken Andrew - I was just thinking fast growing, dense, neat and amenable to heavy clipping. But you're right,it doesn't get much above five feet.

Thanks to everyone for the suggestions.
How fast do hornbeam and beech grow?

Consider, too, the portugal laurel (evergreen), classier than its cousin, with dark reddish petioles, and steadier in growth, but ultimately a better, denser hedge. Beech slightly slower than hornbeam, better on lighter soils, whereas hornbeam is great on clay. My hornbeam hedge produces two to three feet of growth, much of it sideways, each year.