That's an excellent and increasingly important question. The short answer is: It depends heavily on the specific type of plastic and your local recycling facilities. Unlike a standard PET water bottle, the plastic used on public trash bins is designed for durability and weather resistance, which complicates recycling.
Most urban trash bins are made from high-density polyethylene (HDPE) or polypropylene (PP), which are technically recyclable plastics (#2 and #5). However, they are thick, often contaminated, and may contain metal parts, dyes, or additives for UV resistance. This makes them unsuitable for curbside single-stream recycling bins.
At the end of its life, a municipal bin's fate is typically determined by the public works department. Many cities have specialized contracts where damaged bins are collected, cleaned, shredded, and recycled into new plastic products like park benches or, indeed, new bins—a process called closed-loop recycling. Other times, they may be downcycled into plastic lumber.
As a resident, you should never place a broken bin piece in your home recycling. Instead, report damaged public bins to your city's sanitation department. For personal outdoor bins, check with your waste hauler for specific take-back or bulk item recycling instructions. The key is that recycling these items requires a dedicated, industrial-scale process, not the standard municipal collection.