You're wise to forget the wood fence, especially along the canal. My first bit of advice is to run, don't walk, to nurseries and buy Lady Banks roses for that stretch of chainlink fence. At least two of us - me and Mary McP - are big fans of LBR's for privacy. The big plus is that they will also create an impenetrable barrier between you and mischief-makers. They do need some pruning now and then, mostly to remove suckers, but once they are established, the most they need is water - and not a lot of that.

As for the section between you and your neighbor (I'm assuming this is a side fence and you actually have a neighbor there.), I'd choose other shrubs or vines. There is a huge selection of vines that will happily cover the chainlink and provide some greenery and privacy. Your challenge will be to choose one that won't invade your and your neighbor's yard.

Here is a link that might be useful: All about vines

There's currently another thread on the same topic going on here:

Here is a link that might be useful: Privacy Hedge

tteeple: I can't recommend specific plants because I don't know. Desert gardening is fairly new to me. But, having paid someone to landscape my backyard my advice would be to do it yourself if you enjoy gardening at all. Irrigation is pretty straightforward once you do a little bit of it and I've learned more about caring for the plants from the ones I've had to replace (because I didn't know anything about them including how to take care of them) than I ever would have learned from having someone else do the planting. In addition, doing it yourself is much less expensive and the gradual nature of it allows you to figure out what you like and what works.

tteeple, where did you go? Come back and fill us in on your progress!

I have Wild Tamerind, LEUCAENA LEUCOCEPHALA, It is a tree/bush. I really don't like it but it grows like a weed and provides a lot of privacy. Plus you can eat the pods.