This list tells which vines to grow on a trellis:

http://www.hintsandthings.co.uk/garden/climbers.htm

Here is a link that might be useful: Trellis vines

The wooden or PVC slatted diagonal trellis is a more difficult surface to get vines to grow on, since many of the twining vines such as clematis (leaf stems do the twining) or sweet peas (tendrils do the twining) can't readily wrap around such wide supports.

-Ivy clings with adventitious roots and so is able to grow up a surface like your trellis. The ivy may need some guidance to get started on the trellis. Have you tried guiding some of the plant stems up onto the trellis and loosely tying them there to let it get started up the support?
-Is one side of the trellis sunnier? The ivy may be seeking the right light levels and some parts of the trellis don't make it happy.
Be aware that in some parts of the US ivy is invasive and will spread to woodlands and both interfere with native plant communities and pull down trees.

I wouldn't try growing trumpet vine on this type of a structure since the trumpet vine (Campsis radicans) will become too heavy and damage the trellis.

Vines with sticky disk-type grippers such as Virginia creeper (Parthenocissus quinquefolia) or its variegated form (Parthenocissus quinquefolia Variegata) or its more restrained variegated cousin, silvervein creeper (Parthenocissus henryana) should cling to the trellis. Boston ivy is another cousin (Parthenocissus tricuspidata) and climbs the same way. All have lovely bright fall color but can be large vines.

Alternatively, you can put a wire mesh in front of the trellis or fasten wires or hang wires in front of it to allow twining vines like clematis, sweet peas or the native honeysuckle, Lonicera sempervirens, to climb in front of it.

The inside of the trellised enclosure is sunnier...the trumpet vine seems to stay on the shaded side while the ivy covers the sunny side.

The Southwest side of the wall is shaded by bushes.

Are there any twining vines that are good at twining around relatively large things (1 inch)? What about beans?

I've tried sort of weaving ivy through the slats, but most of the ivy's growth still focuses on the ground.