When we request each other to provide evidence in an argument, should we be referring to source papers that contain original data, or is referring to authorities good enough?
If you are citing specific research, then you should always try to cite the original peer reviewed paper that the research is found in. If you don't understand the material, a good alternative is a quote from one of the scientists in a secondary article, like those found in Scientific American. In those articles, the scientists will "dumb down" the findings for a more general audience.
For general knowledge, a well respected college level textbook is a good place to start, and can usually be found in your local library. If you are talking about biochemistry and you quote Stryer, then I know it is probably solid info. Stryer's textbook is still considered to be one of the standards.
If you lack that reference, then your best bet is to qualify your statements, and ask if an expert in the discussion can verify what you are saying.
if I'm not a nerdy book-worm whose primary interest is learning about the natural world, how do I participate in your democracy?
By supporting nerdy book-worms who do study the natural world. Investment in science is an investment in society.
Or more generally, how do you expect a general public who is not necessarily compelled by knowledge to interact with you?
Science and scientists should always be held accountable by the general public. This is done through oversight and making sure that safeguards are in place. For example, Congress makes sure that administrators watch over how public research money is spent, and that research adheres to the ethics set forth by Congress in law (i.e. HIPAA rules for human subjects). As far as the science itself, it is inherent competition in science that keeps things honest. I think we expect the public to leave that to the scientists.