The reason why conspiracy sites like Shadowstats can be proven to be wrong, and government figures can be more or less proven to be (generally) correct, is that there are plenty of
non-government data sources one can use to corroborate the government stats.
A perfect example are the ADP monthly job figures. ADP is a check-cutting firm (largest in the country) which processes tens (hundreds?) of thousands of private sector paychecks every month. They use their information to come up with a monthly estimate of job gains/losses, two days before the government stats come out. Now, on a month-to-month basis the two frequently don't match up (there's a lot of noise in both data sets), but when one looks at the broader trends they match up pretty darn well:
sourceThe conspiracy theorists conveniently ignore non-government stats like ADP because if they didn't ignore them, they would be forced to confront the fact that they more-or-less agree with the "official" government stats. On occasion a doomer type will acknowledge their existence, but since the data they show does not match up with their preconceived notions of what they think/want to be going on in the economy, they then expand their conspiracy to include the private sector entities as well. For example, I can show you a lot of doomer types who think the National Association of Realtors is in on the conspiracy as well.
This tactic, of course, is intellectually honest, to say the least. The doomer types who engage in this kind of thinking are not actually looking for facts, they are deliberately engaging in self-deception (and trying to deceive others in the process). They pick and choose whatever they want to believe, and when something comes along that challenges these beliefs, they engage in all kinds of mental masturbation to convince themselves that what they're being shown is some sort of lie.