OK – for US tax purposes, there are two approaches commonly used for foreign pensions. Technically speaking, the tax treaty says you should only be taxed by the country of origin – so French national pensions (i.e. mandatory pension schemes) are only taxable by France. The mechanism for doing this is a little odd, though. Officially, you are expected to declare all of your foreign pensions on your US returns – and then claim a refund of taxes paid to France on the pensions using Form 1116 (Foreign Tax Credit). If this is your main source of income, chances are you don’t owe any US taxes.
FTC forms, however, can be a real headache to fill out, as you have to break down your income by “type” and then apply only as for the tax credit. Basically, if your foreign income is half of your total worldwide income, you can only deduct half of the French taxes paid against any US tax liability. However, if you really live on your pensions, you should be able to deduct everything you pay in France. The downside to all of this is that there’s a reasonable chance you’ll pay far less tax in France on your pensions than you accrue in the US on pensions.
What many expats do is simply not declare their French pensions at all and leave it at that. The French government certainly doesn’t report anything to the IRS on how much it pays you in pensions, and given the deplorable state of the IRS right now, it’s highly unlikely that anyone will bother you. Traditionally, the IRS has had little or no enforcement power outside of the United States, unless you hit its radar as very high earners (like Boris Johnson).
If you have a copy of the tax treaty handy, check Article 18, which is the one that deals with pensions.