So, after talking with some folks who should know, the answer for US residents comes down to a question of state law. For example, Illinois requires that reisdents self-report a "use tax" for goods as part of their annual return where
- the items bought are taxable in Illinois,
- the items bought are used or consumed in Illinois, and
- when the items are purchased, you either did not pay any sales tax to the seller, or paid sales tax at less than Illinois Use Tax rates of 6.25 percent for general merchandise and 1 percent for food and drugs.
So, mileage may vary by state, but for many states, you would be obligated to self-report your purchase on your annual tax returns and then pay a state tax rate; in Illinois's example, 6.25%, which is at least far less than VAT.
So this is similar to Amazon's tax, before they were compelled by states to collect it. Before, users were supposed to self-report the tax on purchases and pay it on their own. Of course no one did, and it was impossible to enforce.
If that's true, then this is basically the same. We may be obligated, but it is impossible to enforce, and basically no one does it.