Short answer: it depends on the bench. A well-built outdoor bench made from weather-resistant materials can usually stay outside all winter, even with snow on it. A cheap or untreated bench? You are much better off bringing that inside if you want it to last.
Snow is not just cold water. It sits on the surface, melts slowly, refreezes, and works its way into every tiny crack. That freeze-thaw cycle is what really does the damage. For wood, it causes splitting and warping. For metal, it leads to rust, especially if the paint or powder coating has any chips. Road salt and de-icer make things even worse because they speed up corrosion.
Here is how I would break it down by material.
Teak, cedar, and other naturally weather-resistant woods can handle snow quite well. They are dense, oily, or rot-resistant, so they do not soak up water the way pine or spruce does. If your bench is teak or cedar, leaving it outside is generally fine. It will turn gray over time, but that is cosmetic. If you want to keep the original color, cover it or store it.
Pine, fir, spruce, or any cheap softwood bench should not sit in snow all winter unless it is very well sealed. Even then, I would cover it or move it into a shed or garage. If you cannot bring it inside, at least get a good breathable cover.
Powder-coated aluminum and stainless steel are the easiest. They can handle snow, rain, and cold without much trouble. Wrought iron can also survive outside, but only if the paint is intact. If you see rust spots, sand them, touch them up, and then decide whether to cover it. Cast iron is tougher but can still rust through over many seasons if water sits in joints.
Plastic, resin, and poly lumber benches are the modern low-maintenance option. High-density poly lumber can stay outside in snow without issue. Cheap injection-molded plastic may become brittle in freezing temperatures and crack, especially if someone sits on it during a cold snap.
Concrete and stone benches are fine outdoors, but they are heavy enough that you probably were not planning to move them anyway.
If you decide to leave your bench outside, do a few things first. Clean it and let it dry. Tighten any bolts or screws. Apply a fresh coat of wood sealer or metal wax if needed. Remove any cushions or fabric. Then cover it with a breathable outdoor furniture cover. Do not wrap it tightly in plastic sheeting, because trapped moisture causes mold and rust. If possible, raise the feet slightly off wet ground using pavers or small blocks so the legs are not standing in melting snow for weeks.
The only time I would absolutely bring a bench inside is if it is antique, handmade, unfinished, or has upholstered parts. Wicker and natural rattan benches also belong indoors or in a dry shed because they will absorb water and fall apart.
So yes, a good street bench can stay outside in the snow. The key is knowing what it is made of and giving it a little protection before the first storm hits. If in doubt, cover it or bring it in. A few minutes of winter prep can add years to the life of the bench.