There have been some groups doing that. There can be problems with it, though.
Hypothetical situation...
For instance, an untrained group is out searching. They find something, like say an article of clothing, that looks like something Savannah wore that night. Because they're not trained in the handling of evidence, they walk all over the scene, potentially destroying any footprints or tire treads that may be there. Then they pick the article of clothing up with their bare hands, getting transfer DNA on it and potentially shaking off any hairs or other smaller traces of evidence that may exist. Lastly, they remove the article from the scene without taking a picture of it the way that it was found so LE has no idea how it looked when it was discovered. Then, after all that happens, they go on social media with photos of them holding it, talking about how they found "evidence". The family see it and it generates a lot of upset. In the end, it winds up not being anything at all. Then again, it might really be something, but in court it's thrown out because of the way it was handled.
This is why volunteer search groups need to be trained and organized and why groups of people can't just go out on their own. Things like this happen a lot. It's why we ask that, if a group wants to search, they meet with an official SAR official and have someone speak to the group before they set out.
Now, with all that said, we do need more searches. We do need more organized group efforts. There are ways to do that, however, that are mindful of evidence.