Short answer: yes, you can paint a street bench after purchase. You do not always need an exotic specialty primer, but you do need the right primer for the material—and prep is even more important than the primer itself.
Most benches you buy are either wood, metal with a factory finish, or some kind of plastic or recycled plastic. Each one behaves differently.
Wood benches: If the bench is bare wood or only lightly sealed, start by cleaning with a mild degreaser or soapy water, rinse, let it dry, and sand with 120- to 150-grit paper. Use an exterior wood primer, especially if the wood is cedar, pine, or fir. If the bench has an old varnish or glossy sealer, scuff it well and use a stain-blocking exterior primer. Then apply two thin coats of exterior acrylic latex paint. For wood bench painting, thin coats beat one heavy coat.
Metal benches: A brand-new metal bench often has a powder-coated or factory enamel finish. Paint can stick if you scuff-sand the surface with fine grit, wipe it with mineral spirits or denatured alcohol, and use a bonding primer or direct-to-metal primer. If you see bare steel or rust, use a rust-inhibiting metal primer first. After priming, spray or roll on a direct-to-metal enamel. That metal bench paint will hold up much better than ordinary wall paint.
Plastic or recycled plastic benches: This is where adhesion is trickiest. Clean thoroughly, scuff with a fine abrasive pad, and use a plastic adhesion promoter or plastic primer before painting. Top with an exterior-grade spray paint made for plastic or a high-quality acrylic enamel.
General steps: clean, sand or scuff, wipe down, prime, apply two thin topcoats, and let the bench cure at least 24–48 hours before heavy use.
Do you need a special primer? Yes and no. The primer should match the bench material—wood, metal, or plastic—but it does not have to be a boutique product. A good exterior or bonding primer from a hardware store is usually enough. Skipping primer is the biggest reason fresh paint peels off a store-bought bench.
One last tip: if you are painting outdoor furniture that will sit in sun and rain, choose a paint labeled for exterior use. Even the best primer cannot save interior latex from weather.