I am assuming you realize that elephant ears can be extremely invasive in the south. They like shade (though will grow in full sun), abundant moisture, and nitrogen. They are not evergreen, so they will not hide your fence in the winter.

If, after these considerations, they are still what you want, plant them about two feet from the fence and three to four feet apart. Their running roots will fill in very very solidly by next year. (Be sure your neighbors won't resent them running into their yard, because they most certainly will do that.) Think this decision through very carefully. Read some other threads about elephant ears, and know what you are getting into, because they are extremely difficult to eradicate. I know from experience.

If you decide you want something else (you do, don't you?), I would recommend a nice, loose, natural hedge of evergreen shrubs. You could choose one type of shrub for a traditional look: hollies, ligustrums, cleyera (my favorite), loropetalums, just to name a few. Or, you could plant sweeps of 5 to 9 shrubs of one type together, and have different kinds of interest coming and going thorughout the year: for instance, a group of camellias (late winter/early spring bloomer) followed by a group of large azaleas (later spring bloom), followed by a group of roses (not evergreen, but mighty gorgeous all summer: the new Double Knockout Red is outstanding), followed by a group of Oakleaf Hydrangeas (early summer bloom, beautiful fall/winter foliage).....etc. See? If this is too much money to spend all at once, just buy a group of shrubs each year and gradually add to the border all the way around your yard. Plant them the correct distance from one another (as indicated on their tags), and half of that distance from your fence. For an even nicer look and better growth, dig out the grass about four to six feet from your fence all the way around, make a nice neat edge to your lawn, and then mulch. Your yard will look like a million bucks. Put your elephant ears in pots! :)

I would also recommend putting the elephant ears in pots. I spent much of my childhood and teens trying to get rid of the elephant ears my mother planted along the front of her house. Eventually, they had someone "remove" the elephant ears. However, 30 years later, they still occasionally come up.

Also, consider carefully before planting ligustrums. If you have allergies or asthma, there will be a significant period of time each year that you will not be able to be outside in your yard.