For instance, one of the new trends in prediction markets has been corporate elicitation (e.g., Inkling Markets YC W06). For instance, a market might ask a pool of employees "When will product X ship?". Now consider the employee's problem. He knows the product will be delayed until November at the current pace, but that if managers see high November probabilities they'll cut back on the project and re-focus development efforts (or, perhaps instead, shift some staffers over to get the project done faster). How should the employee trade in this market?
If your answer is "it's unclear" then you're exactly right. The self-referentiality of prices (they cause actions which then define prices) makes the problem much harder. It's actually possible to design markets which are so self-referential that any trade will be the right one (the market prices are self-predicting prophecies) or that any trade will be the wrong one (e.g., if there's a high price on terrorist attack Z then the government will always spend enough money to prevent Z from happening).
In fact, the only way around the paradox is to have the prices in a prediction market not matter at all - for the managers to take the same course of action regardless of the prices they see in the market. But then why run a market in the first place?
The whole thing is an interesting and seemingly fundamental perversion of the way we normally think about prices. And again, this is without any malicious intent on the part of participants, which will certainly only further confuse things.