First, unless you and your neighbors actually dislike each other, talk with them and see what they think of the existing fence. It may be that y'all can agree to change out the entire fence and they pay half of the part that they see.

If you must do it alone, you have choices.

BEST BOARD OPTION: If you choose boards, nailing the less expensive
grade of 1"x6" deck boards horizontally will actually be more sturdy
even with less posts than using thin boards nailed vertically -- and the
rounded edges would shed water better than the flat edges -- better for
the wood.

If the posts are on your side of the common fence, you could add posts between those posts and affix your boards to the new posts but not the old ones. With your own new posts, you can build whatever kind of wall you want on
those posts -- preferably the same on all sides -- and you can then let
your neighbors repair or replace the existing fence boards they see
when they decide what they see needs work.

Even if the posts are not on your side of the fence, it may well be that foregoing two and a half inches of yard beneath the shared fence and building a whole
new fence in front of the existing fence ends up being the easier
and, in the long run, least costly option. Then, the entire fence would be yours to change again in a few
decades if you decide to do so. You'd need to use two posts to create your own corner rather than a single corner post.

If you choose a chain link fence, you may not be able to build as close to the existing fence -- get a quote and info about this from a fence contractor.

After you get your new posts in place, create a paving stone or brick line with the tops of the stones level with the yard between the posts or just pour a concrete curb to discourage animals from digging beneath the fence and so you do not need to have your new fence actually touch the dirt. You can then lay a square treated post horizontally atop the paving stones and attach them to your post with long coated deck screws and build your fence so the bottom of the wood is just barely above that. A decade or more in the future you might eventually need to change out that horizontal post atop the paving stones while the fence itself, if built with deck boards, is likely to still be in good shape.

Coat all posts and boards with CWF on all sides and edges and ends before placing the boards.

Edit to add: just realized this thread began in 2013 -- what did y'all decide?

Trex put out a horizontal fence last year. It's low-maintenance. Won't need painting or staining and has a 25 year warranty. It's not made from their deck boards but it is composite with a horzontal design. You can find out more at trexfencing.com/horizons.