When I was lived in the US, I received W-2's from my employer, and 1099-INT's from the bank where I stored my money.
Now I'm living abroad, and I am not getting either W-2's or 1099-INT's. However, I still have an employer and make money from interest. Do I need to file some form to report these to the IRS, so they would know which employer paid how much, and how much of my income is interest? Publication 54, Guide for US citizens living abroad, seems to offer no information regarding this.
The W2, 1099, and other forms are informational and are used for matching. They do not determine what your taxable income is or how much taxes you owe.
So receiving or not receiving an informational form doesn't matter to your reporting, as a US citizen you should report all your world-wide income in the same manner.
Specifically for foreign earned income there are some additional notes:
Salaries - since you didn't get a W2, you don't want to report the salaries on line 1a, instead report on line 1h. This helps avoiding matching errors with the IRS computer.
Salaries, again - if your employer is an affiliate of a US company, there may be withholding for FICA taxes (see here). You might want this to accrue US SSA benefits if there's no totalization agreement with your home country (and your employer can do this).
Salaries (and any other earned income) third time - Don't forget about the Foreign Earned Income Exclusion, that allows you excluding some (all?) of your foreign earned income from US taxation. Note, this only applies to earned income (salaries/contract work), not other income like interest or capital gains.
Foreign tax credit - if you paid income tax on your income in the foreign country, you may claim some (all?) of it as a credit towards your US tax liability. You cannot double dip, though, so you can't claim credit for taxes paid on the income that you excluded using the FEIE described above. Note, the "income tax" is defined by the IRS, and additional taxes imposed on income in the foreign countries may qualify for this credit.
See also the instructions to the form 1040, and the Publication 54 (HTML version) you already mentioned.
You tagged the question with "us-citizens", so I'm assuming you're a US citizen? Foreign income would generally be reported the same way as US income. For example, foreign interest would go on the interest line, foreign capital gains would go on the capital gains line, foreign rental income would go on the capital gains line, etc. You don't need to have received a 1099 form to report it (e.g. you don't receive a 1099-INT for interest of less than $10, but you still have to report it; you don't receive a 1099-NEC for payments of less than $600, but you still have to report it).
For wages, instead of line 1a, "Total amount from Form(s) W-2, box 1", if your foreign income doesn't come on a W-2, you would instead put it in line 1h, "Other earned income".
