Here in the UK the vast majority has very simple income (i.e. just salaries). The employer reports the income and taxes taxes and passes it directly to HMRC (our IRS equivalent). By the end of the year for the vast majority HMRC knows how much you've paid in taxes and how much you've earned, and automatically gives you a rebate in case you've overpaid, or adjusts your tax code next your to pay what you still owe (if it's not a big amount).
If you have additional income, dividends, gains and under certain other circumstance, you're liable to fill out a tax return (self-assessment as it's called). It's a very simple form that contains information that applies to the vast majority of those who need to do their self-assessment. After you've sent it back, they'll calculate if you've overpaid (and compensate) or underpaid (and require you to pay what you owe) or neither. It is trust based but they do audits up to 7 years back, and if have lied or made mistakes, then you'll be liable.
The few richest of the richest who fall outside the above two (they still need to do a self-assessment) probably have a completely different process but it doesn't matter because they have the resources to hire firms to figure it out.