I primarily study political conflict between crown and assembly in Jamaica and New York in the period 1675-1740, so I may not be working with the kind of newspapers you're interested in, OP. That said, there were newspapers in New York City, Kingston, and London in my period that I do use as sources, so hopefully I can still shed some light.

The first answer, which is a bit of a dodge, is that often I don't care all that much about whether the story is factually accurate. I'm often more concerned with what the story tells me about the writer's agenda and worldview, and the concerns of the anticipated audience. This is often a much safer way to approach my newspaper sources, since the ideal of unbiased journalism was still a long way off in my period.

But let's assume I really do care about whether a story I read is factually accurate. The second answer, then, is that I'm usually triangulating. It's rare for any important event or controversy about which I'm reading not to be mentioned in other sources. So I'll check rival newspapers, governors' correspondence, journals of the assembly, letters of other prominent people, and so forth. All of these will invariably carry their own biases, often in quite ludicrous form. So I assume that they are all distorting and massaging the truth for their own benefit, and I try to average those likely distortions out to a plausible construction of events. I can never be wholly confident that I've got it exactly right, but I'm generally confident that I've found the best fit. I tend to be happier with that best fit, by the way, the more messy and complicated it is, as a general rule. Simple stories feel implausible, because any one person will have multiple motivations for taking any important action, and adding more people will similarly add to the complexity.

As a side note, if I'm lucky, something in the story can be corroborated with government-generated statistics, which for my work tends only to be the case if the revenue is at issue. Eighteenth-century statistics aren't the best or the most reliable, but they really do help a researcher nail down whether these Jamaican planters really had any right to be protesting a downturn in trade in 1715 as justification for not quartering soldiers.