My experience with pyracantha is that it has exactly as many thorns as leaves, and pretty much grows as a tree. I've got one woven into a trellis but it's not really paying attention to where I want it to be. It just grows. Definitely a good anti-climber plant.
Neighbour has winter jasmine growing up the wall and a trellis on top of the wall and trailing down on the public side ... two feet thick. Fortunately, it's a wide path. That's not really a climber, either, just a bush that tends to be taller if it has somewhere to rest on the way up.
Honeysuckle Lonicera and Chocolate Vine Akebia Quinata would be my first suggestions, because my experience of clematis is that they have a 12-month half-life. Of the two, Honeysuckle definitely favours the sunny side, so if you've got a south-facing fence there you'll probably be looking at the leafless inner branches while the public get the display. You could probably beat this by having fence panels on the public side and trellis on your side. ChocVin's a bit more inclined to just grow in all directions, but really likes being sunlit.
I've got "climbing" roses too, but attempts to get them to branch out into big fans of climbing roses merely caused the root stock to grow past the grafted climber. Turns out the root stock produces gorgeous flowers, so I'm letting it do that. Again, though, it's not really doing the trellis thing.
If it's a long fence, I'd suggest putting purple and white varieties of Akebia and a few http://www.bluebellnursery.com/cgi-bin/search.cgivarities of Lonicera along it, pruning them each spring for a few years to get them to spread out and then letting them mingle on the trellis. It'll be a few years before you get really good cover, but you should get a charming mix of foliage and flowers and loads of hoverflies, your very own anti-aphid air cavalry! A little layering will spread each plant both ways, to give you a more thorough mixture.
Just beware with the Lonicera that not all of them are climbers. Some are shrubs and some are best grown as hedges.
http://www.bluebellnursery.com/catalogue/climbers/Berberidopsis/B/293Berberidopsis I don't know, personally, because it might not like our winters down here in the frost hollow, but if you get the trellis filled with other climbers it should cope with growing up through them.
Having typed all that I read the rest of their list and hit Wisteria at the bottom. Hmm. Well, you'd certainly get some privacy with one of those 8m high and 8m wide but you might need planning permission for that trellis. O.o
If you like it riotous, the mixed honeysuckles and chocolate vines with berberidopsis to follow once they provide shelter. If you like it neat and simple, just one or two kinds of wisteria.