[gravitas]

Both. Financial institutions (broadening this to cover brokerages as well) produce federally required documentation (it may be digital in 2020, automated) for tax purposes; the most common in my travels is a 1099-DIV (dividend payments) but there are many kinds for all sorts of financial things a bank could/can handle related to investment income.

Both parties submit paperwork to the IRS - the institution tracks the assets they're holding/managing and reports it to the IRS on their side; they issue you appropriate documents and you do your taxes and report it using whatever appropriate form. The paperwork is of course dependent on the situation, for the average checking/savings accounts I will guess it's the 1099-DIV as most common - simple interest payments.

[gravitas]

I got that slightly wrong, it's 1099-INT for interest in income on checking/savings, 1099-DIV is for dividends only.