Earlier quoted context omitted.
The problem with a person doing this style of tax avoidance is that the barrier to entrance is much higher than the gain. Assuming general principals of US tax law, if you're already working as an independent contractor, you could legally set up a local company and an overseas company, have the local company bill the client, pay you a reasonable amount, and pay the overseas company the remainder for the use of its na…
The scheme you are suggesting is money laundering. The reason it might work for some big names is that those have stronger reasons to bill the off-shore jurisdictions. For example, a Starbucks LLC in "NewTown","Small-EU-Country" is fine billing an "off-shore" company for the use of Starbucks name. The same is not fine if you are opening a local coffee shop.
Different countries have different laws to combat the use of foreign companies, though. It might end up not being worth it because there are other costs associated with setting up an entity (e.g. some substance requirements)