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Yes, you can repaint them. First, wash them thoroughly with a detergent that contains a bleach or acid-based mildew killer to remove all traces of both dirt and mildew. There are various premixed products available at your local hardware store that will do the job, or you can use TSP and add bleach to it. Allow the shutters to dry and then lightly sand them with a 120 grit sandpaper to slightly rough up the surface for better paint adhesion.
An exterior latex paint is the only choice for your finish coat. Latex expands and contracts quite a bit, so the paint must be flexible enough to stretch, too. Applying a latex primer is not absolutely necessary but it is a nice security blanket and will undoubtedly help the finish paint to stick even better. Generally, primers are used to even up the absorption of the finish coat by sealing the surface. Since vinyl is not porous this "prime"ary function of the primer is not needed. However, it should be noted that primers cling more tenaciously to surfaces than finish paints. So if you don't mind the extra work, priming first is not a bad idea.
Two other things... it can be helpful to tint the primer to match the finish paint, especially if you are making a significant color change. Also, have the paint store add a package of exterior-grade mildewcide to the finish paint. Unfortunately, most modern paints and stains do not resist mildew very well "out of the can".
Be careful with your cleaning methods if you plan on painting the shutters while on the house! The cleaning process can leave streaks on the siding from the cleaning product runoff. Your choices are to take the shutters down, clean the entire house or get the siding very wet before using the cleaner so that there is less likelihood of drip marks.
There is some debate as to whether vinyl siding should be painted. The controversy surrounds the expansion and contraction of vinyl siding, which is much greater than wood or aluminum siding. Paints that do not expand with the siding will crack.
However, this is not a problem if you use top quality acrylic latex primers and paints. Acrylic latex paints are remarkably flexible and, providing the surface is thoroughly cleaned, will work well on vinyl.
The siding should be cleaned and dried thoroughly. For hand washing, Owens Corning recommends the following cleaning solution: 1/3 cup powdered detergent (Fab©, Tide© or equivalent powdered detergent), 2/3 cup Spic & Span© or equivalent powdered household cleaner dissolved in 1 gallon water. If your vinyl has mildew, substitute a quart of household bleach for one quart of water in the mix.
Apply the solution with a sponge or an automobile brush. Rinse thoroughly while still wet.
Of course, you could simplify your life and rent power washing equipment, which would be quicker and also make cleaning the upper level somewhat easier!! Just be sure to get a cleaning solution designed for powerwashers... or you might end up loosing you home in a mountain of soap suds!!
This is a letter NH wrote to Krylon concerning their new spray paint "Fusion"...
Thank you for your inquiry with Krylon. "Fusion" can be used for both surfaces provided you use a lighter color then that of the surface. Using a darker color will absorb more heat and will cause the surface to buckle.
Though "Fusion" will bond to vinyl siding, the cost per can does not justify its use for large surfaces. The Fusion is not available in brush on formula at this time. I suggest using bucket latex paint for the siding.