Depending on the scissors, yes, scissors can be sharpened, although they generally require the attention of a skilled sharpener. It is also possible for scissors to be sharpened at home, if you are comfortable with a whetstone. When you do sharpen scissors, you may want to think about the value of the scissors, as high quality scissors are worth the effort, while cheaper scissors may be better off being discarded.
When scissors are sharpened, the beveled edge on one side of each blade is renewed, refining the edge of the scissors so that it is sharp again. Scissors do need to be sharpened periodically as even routine use will eventually dull them, and you want them to have a crisp cutting edge. You have a number of options when it comes to sharpening scissors, depending on the manufacturer. Some manufacturers will allow consumers to mail in their scissors for sharpening, typically manufacturers of high end scissors. It is also possible to hire a knife sharpener to sharpen them, or to sharpen scissors yourself at home.
If your scissors are eligible for sharpening by the manufacturer, you will generally be given information about that at the time of purchase. Buying high quality scissors is a good idea for a variety of reasons in addition to sharpening support; they tend to hold up better, for example, over the long term. Typically you must sheath the scissors for mailing and wrap them in plastic wrap; some companies will even send you a mailing envelope. Depending on the manufacturer, you may be asked to pay a small fee to sharpen scissors.
If you cannot send scissors out to be sharpened or you don't want to, you can approach a professional knife sharpener. Not all knife sharpeners sharpen scissors, so make sure to ask. As long as you're sharpening your scissors professionally, you may as well take all of your knives in, establishing a routine which ensures that all of your blades stay in good condition. A scissors sharpener may also have suggestions for extending the life of your blades.
To use a whetstone to sharpen scissors at home, you merely need to ensure that the angle on the blade is matched by the angle of the whetstone, as you want to crisp up the edge, not refine a new one. Move from the end of the blades toward the central pivot with smooth strokes of the whetstone until the blade is sharp, repeat the process on the other blade, and then run the whetstone lightly along the back of the blade to remove any burrs.