The potential for regular people contributing to science is great. This has a long history. For most of human history science was done by non-scientists since there were no scientists. Calling is science might be a stretch but to me it was (passing on what health cures worked for various sicknesses, how to use various tools, how to grow crops…). As scientists came into being they were primarily unprofessional – that is they practiced science but were doing so as a hobby, they were not paid and had no requirements to get a PhD or anything.
Today regular people help by collecting data (counting birds, documenting plant growth [time of year], migration data, weather data…) sharing knowledge with scientists who ask, sharing their computer to be used to analyze data, analyzing data (for example, in astronomy hobbyists often make new discoveries) and the latest way people help is through games (that essentially tap human brainpower to analyze data – such as Foldit, which I have posted about previously).
I like the contributions people can make to science but I think the biggest value is the scientific understanding people gain while participating. As Neil Degrasse Tyson says the scientifically literate see a different world.
Cornell University provides an online tool to find opporunities participate in scientific research.
And we shouldn’t forget the amazing science done by students like those honored with Intel Talent Search, though the work those winning the awards do I would lump with science by “real scientists” (I believe now most of those who win are working on projects with university scientists).