I am pretty good at talking to people, so my first step would be to carefully talk to the neighbours. Pretend to like them. Pretend to be interested in something they like (e.g. buy some big dog chew toys for their dogs & tell them you think their dogs are fantastic). Propose a privacy scheme that involves/includes them. Offer to plant something (for free) on their side of the border that might interest them.
You can buy inexpensive, dog-proof fence quite cheaply in the United States. 4in. by 4in. or 4in. by 2in. fencing, in 100ft. lengths (48in. tall), is $100-$200, probably. You can use that fencing to seal the property while you pursue a long-term solution. The Trifoliate Orange 'Black Dragon' cultivar is most certainly dog-proof -you do not need the 'Black Dragon' cultivar, it is just the most horrifically-thorned of the species. A thicket of native/wild (thorned) raspberries is dog-proof. Other options: Bamboo
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Slightly related:
There is a terrifice podcast called 'The Dollop'. They have an episode about a famous "spite fence" in San Francisco:
http://thedollop.libsyn.com/70-the-spite-fence
