I like those sharpeners you get at Home Depot and a lot of other places for general sharpening, but my good shears, no! Try a Singer store, or JoAnn's Fabrics.
Joanns has a guy that comes in periodically. He sharpens both scissors and knives. Call and they can tell you when he's scheduled to come in. It is realatively expensive, so I use my coupon and buy a decent new pair. Disposable! It's cheaper.
I'm pretty sure that Jim's Vacuum and Sewing in Panama City does this, but if not, the JoAnn Fabrics in Panama City has GREAT prices on Ghinger shears as well as really nice rotary cutters, and the ladies that work there are very helpful with advice on supplies!