A copyright symbol used to be required in the past, but—since March 1, 1989—it’s no longer required (an effect of the Berne Convention Implementation Act, Copyright Law). The moment you press the round camera button on your iPhone, draw a doodle on your iPad, or snap a photograph with your digital camera—it’s yours—and it’s copyrighted. From that very moment, you are the only one who has the legal right to reproduce or publish your work (Merriam-Webster).
According to Section 504 of the Copyright Act, even without a copyright symbol, an image is still subject to copyright law. In fact, if you decide to use an image without permission, you could pay anywhere from $750 to $30,000—not to mention other court fees.
However, if a copyright symbol or watermark is present and you infringe upon that work, it’s considered intentional and you could pay upward to $150,000 (per infringement) in court—yikes!
Some sites such as Creative Commons allow users to search for images from owners that have given some level of approval to use, build upon, or share their images. These often require attribution, linking back, and permission. However, It’s probably best to assume all images are copyrighted—symbol or not.