Advocates of prediction markets often focus their attention on markets that can be run for profit. This may be due to a bias resulting from the fact the prediction markets mainly interest people who are trying to combat prejudices against profit-oriented activities.
Profits are often better measurements of whether something is valuable than the average person realizes, but there is little reason to believe this applies to public goods. And much of the motivation behind prediction markets advocacy is the need to make better information available to voters about the effects of policy choices. That information is about as clear an example of a public good as one can get.
An assumption that a for-profit corporation is the best way to produce a public good should normally be treated with some suspicion (although alternative types of institutions deserve suspicion as well).

A for-profit prediction market that tries to both maximize profits and provide valuable information to voters faces conflicting pressures, since my observations of existing markets such as those on Intrade show little evidence of a significant correlation between the issues that create profitable trading volume and the issues on which a democracy needs more knowledge. If maximizing profit doesn’t provide useful evidence that the company is achieving its goal, then it’s unclear why we should think that a profit-oriented form is the best way to achieve that goal.
Automated market makers that transfer money to traders seem to be a promising way to improve the accuracy of market prices, but seem inconsistent with a profit-oriented goal. They look more like an act of charity than a business practice when the goal is to inform voters.
Prediction markets face significant regulatory uncertainties, and governments are typically more lenient about regulating charities than profit-oriented companies. This may help nonprofit prediction markets both because a nonprofit will have less incentive to pressure traders to make unwise or unfair trades (i.e. the concerns behind gambling laws and laws mandating disclosure will be somewhat weaker), and because a nonprofit is less likely to encroach on the businesses provided by existing financial exchanges.
What many of us want from prediction markets bears more resemblance to the products of think-tanks than it does to any other institution that I’m aware of, and the basic ideas behind such uses of prediction markets are accepted by enough people that we can hope some wealthy donors would contribute to a charity that funds such markets. So it seems natural to me that we ought to focus a significant fraction of the thought about how to implement prediction markets into how to create a charitable organization for that purpose or persuade an existing think-tank to adopt that purpose.