What you're touching on, of course, is a couple of basic facts about epistemology — and how they impact the activity of proseletysation: the attempt to get someone to believe in an idea (whether religious or secular) which they not only did not know, but did not even concern themselves with, before.
The short version is that because original discovery of knowledge is difficult, we have systematically done out best to substitute discovery with learning from others. This makes knowledge socially contingent, and therefore based on reputation.
Discovering knowledge is hard. So hard, in fact, that we have developed several interoperating institutions whose purpose is to preserve knowledge which we have won, and to try to disseminate it much more efficiently than it could be independently re-discovered by anyone who might need it. Meanwhile, we pay researchers moderately generous salaries (much less than programmers, medical doctors, and financiers, mind you, but generous considering their outwardly apparent output) to take the risk of brain damage involved in spending long periods of time trying to discover things.
Knowledge of sufficiently complicated topics is socially constructed. The ideas of 'trust', 'faith', and 'teaching' are predicated on the idea that we can take as candidate "facts" things which we have never experienced, but have been told. In this sense, we defer the obligations for knowledge to others — we make knowledge dependent of social relations.
Application to proseletysation. Suppose you want to apprise someone of a new idea. In trying to provide them with 'knowledge' by social means, you must have one of the following two resources: concision, or reputation. If your claim is outlandish, but the explanation is short and sweet, I may humour you just out of curiousity because the cost to me is low. Otherwise, if your explanation is long and complex, I will only investigate it if I have some good reason to give you the benefit of the doubt: for instance, if you are widely recognised by society as a trustworthy speaker on the subject (e.g. you are a famous scientist, you are a teacher at the school and your audience is your class, etc.)
If a conspiracy theorist wants to convince you of something, they should either present it very calmly and soberly (or convince you of short separate pieces of their reasoning which can stand alone) in order to build up your trust in them, or they must provide a very short and compelling explanation of their theory. Otherwise, even if their claims are true, they are asking you to make an unreasonable expenditure of your time and energy as an information-gathering agent for one very specific piece of knowledge — if indeed it is knowledge, i.e. a faithful representation of reality.
This is true not just of conspiracy theorists, but also of religious enthusiasts, and technical enthusiasts — it is unreasonable to ask your friends to share one's excitement for (or vehement rejection of) Jesus Christ as your saviour; and unreasonable to ask one's relatives to develop highly specialized knowledge of the operating system on their computers, if they have computers. There are exceptions in both cases if it is highly relevant to their lives, but if it requires them to spend a lot of attention or to overturn a lot of ideas of how things work, then it will be labour-intensive for them, and one's request should be made giving this fact due respect.
In short, anyone who is asking you to believe in something, is asking you to make an effort. They are requesting a favour from you (the aspects of what makes belief in an idea a 'favour' is itself an interesting idea, but never mind that) in the attention you spend on them. If they want you to do them that favour, the onus is on them to make sure their request is reasonable, in that it is not a social imposition. If they develop a reputation for social impositions, that's unfortunate — but repairable, to the extent that any reputational damage from social transgressions can be repaired.
On noisy learning environments. On the case of penicillin and the Pasteur Institute mentioned by Michael in his earlier answer: we have in that case the unfortunate situation of an environment which was, for the Pasteur Institute, hostile to learning, in that it was subject to a lot of noise by (possibly even well-meaning) cranks. It is not immediately clear what they should have done in the face of such stimulus. There are similar problems in computational complexity and proofs either of P = NP or P ≠ NP: because there are so many well-intentioned (but poorly trained) people who continually attempt to prove it, and who cannot even be induced to learn from their mistakes, the very question has a taint upon it, so that there is a high reputational burden involved in being taken seriously.
Thus we see a second-order ethical obligation to conspiracy theorism (or interests which diverge from the norm, generally): not only should you be respectful of the attention which your audience is literally paying to you, it is your social responsibility to be careful in how you engage in your conspiracy theorism, lest you poison the well for others who may come after you, and thereby inadvertently hurt society.
Again, I think that this responsibility is not just upon conspiracy theorists and amateur scientists.
One of the major reasons why there is as much distaste for mathematics, and wilful innumeracy, in the general population, is because (a) tremendous emphasis is put on the importance of mathematics to make math education mandatory, but (b) it is not taken seriously enough to ensure that the teachers of mathematics are up to the job demanded of them. Students feel their time has been wasted, and conclude that mathematics is likely to be a waste of time. And they then take pleasure out of being part of a majority of people who feel that way.
The reason why such genres of entertainment as comic books, science fiction, etc. have been held in disdain at various points in time, is largely because the well has been poisoned by a small fraction of well-meaning enthusiasts, who ruin the subject by association through their overexuberance.
This is profoundly unfair, but it is better to recognise the fact, because without completely overturning the ways in which information is collected and distributed (and how would one begin to do this without confronting the social nature of knowledge?), it is unclear how one can avoid such problems.
