Answer: There are a number of plants you could use, but please understand that if you cover it completely or enough for privacy you will block the light during the months when the plant has its foliage. Also, it may be that a visual block is insufficient to prevent the dogs being disturbed -- they may hear the passers by and continue to be disturbed by them as a result.
You could consider perennial flowering vines such as clematis, Virginia creeper, trumpet vine, or possibly the red flowered honeysuckles. Keep in mind that vines usually take several years to become established and begin to grow at their typical rate, so the initial coverage will be sparse and slow. Or, you might like inexpensive, quick growing, annual flowering vines such as morning glories although they have to be replanted each year.
To some extent, too, your selection would be determined by the growing conditions along the fence. Most vines do need direct sun to grow well, although the Virginia creeper will do well in shade or sun. Your local professional nursery staff and/or county extension should may have additional or different suggestions based on a more detailed understanding of the growing conditions, the area to be covered, and your design goals.
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