Cleaning and staining a fence with segments of different color and age
The best thing you can do is try and reason with your neighbors. Good fences do make good neighbors and having been in the fence business for 30 years I know how it can be.

Try to explain to your neighbor's that you would like to stain the fence to preserve it. And that you will be paying for the stain and supplying all the labor.

Treating the fence will indeed add many years to the life of the fence. They might consider it if you approached them that way.

But no, you have what is called a shadow box style fence with lattice. Treating one side will not preserve it and there is no way to not get treatment on the other side. It will also look horrible from their side having partially stained areas. Even if your doing it just for aesthetics the same problem will arise. 

If they agree when you use the pressure washer be sure and adjust the flow so the stream does not cut into the wood leaving what will look like scrape marks.

The product I use is Preserveawood. It is found only in Home Depot stores. There are three colors, Redwood, Pacific Redwood which is a bit darker and Cedar. Right next to where they sell the Preservawood they sell those little sprayers like bug sprayers. They put them there for a reason. The sealer comes in 5 gallon buckets. One bucket is good for a 400 lf.' of 6' high fence. Cost is about $125 for 5 gallons, $17.00 for the sprayer. DO NOT get the $10.00 sprayer it will clog up.

Just be sure and mix it well, pour it in the bug sprayer and spray it on. The sprayers come with 3 fan tips that give an even spray. Use gloves because it is oil base. The wood sucks the treatment up. Very easy and effective. It will last for 5 years or more actually.