One simple thing that media entities could do to improve trust is to publicly declare exactly who finances them, and then in turn declare their backers. This would hold true for privately-owned companies; trusts; crowdfunded publications; new kinds of media companies operating on the blockchain and funded with ICOs.
VC-funded media companies - like Facebook, which is a media company - would declare which entities own how much of them. As it happens, Facebook is publicly-traded, so must already do this. But it's rare for VC firms to talk about their Limited Partners - the people and organizations who put money into them. We have no idea who might have an interest in the organizations on Facebook's cap table.
This is important because LPs decide which funds to invest in based on their goals and strategy. It's clear that an LP's financial interests may be represented through a fund that they invest in, but it's equally plausible for their political and other strategic interests to be represented as well.
To be specific, we know that socially-minded LPs invest in double bottom line impact funds that strive to make measurable societal change as well as a financial return. It seems reasonable, then, that some LPs might seek to promote significantly more conservative goals. In the current climate, imagine what a Kremlin-connected Russian oligarch might want to achieve as an LP in a US fund. Or a multinational oil company, the NRA, or In-Q-Tel.
The same goes for crowdfunded ventures. What happens if a contributor to a blockchain-powered media startup is the Chinese government, for example? Or organized criminals? It would be hard to tell from the blockchain itself, but understanding who made significant contributions to a publisher is an important part of assessing its trustworthiness.
While it's fairly easy to figure out which venture firms have invested in a media company, those same firms usually have a duty of privacy to their LPs, so it's rare that we get to know who they are. We know that media is the bedrock of democracy. In order to determine who is shaping the stories we hear that inform how we act as an electorate, I think we need to start following the money - and wearing our influences on our sleeves.