Unfortunately, it is generally NOT allowed in the U.S. Ski resorts either operate on land that they own (rarely), or on public land, with an exclusive use permit usually through the U.S. Forest Service. Either way, they have the exclusive legal right to operate a ski school on "their" land, and no one else can teach skiing there without their express permission.
That does not mean that "under-the-table" private lessons aren't done on the side, of course. They are available. But beware--any instructor who can get away with it is, practically by definition, not one you'd probably want to ski with! Any instructor with any kind of reputation--i.e. any clinician, Examiner, or otherwise high-profile, experienced, "desirable" instructor--will be well-known to the ski school management and many instructors at any resort. There's no way they could "sneak in" and teach without being caught, and few would risk it anyway, as professionals. That leaves only a few undesirables--unknown instructors with probably very little experience, qualifications, or credentials--who might be able to get away with teaching under the table once or twice before they get caught. You'd do better to take a group lesson!
It's not like this in much of Europe, and I agree with you that this anti-competitive environment is a terrible thing for students and instructors alike. With no competition, and no regulation, ski schools can become cash cows for resorts. They name their prices, pay abhorrent wages that don't come close to providing a living in most expensive resort areas, and put horrendously incompetent instructors in front of large groups of skiers. You do NOT get what you pay for, as you've noted, because the great majority of what you pay for an instructor goes instead to some distant shareholder, and to pay for the royally excessive salaries and benefits of a few top executive officers.
Because the industry has taken such selfish advantage of its ability to operate without competition, the quality of typical ski instruction has taken a dive. It's ironic, because the ski industry's own studies have shown, time and again, that ski instruction can make or break the industry. They have recognized that quality instruction turns casual first time skiers into regular devotees of the sport, and takes intermediate and advanced skiers to the next level, keeping their passion alive. Their studies reveal in quite certain terms that the more skillful the skier, the more time and money he or she spends at ski resorts. They task their ski school directors with increasing their penetration and retention/repeat business rates.
Yet they refuse to stop sucking the life out of the very instructors who could provide the key to growing their business!
Anyway, you didn't ask for this rant, Rod, and I apologize. Your question is innocent and reasonable. The basic answer is "no."
On the other hand, there ARE a few alternatives. Our own EpicSki Academy represents one of the best, and it really did break new ground. They aren't private lessons, but they do involve some of the very best instructors, many from outside the host resort, working independently, but with the blessing of, the local ski school. EpicSki Academy is a celebration of great skiing and learning, and enlightened resort management recognizes this as a good thing, even if it doesn't dump money directly into their ski school's coffers. We want to get people excited about the great opportunity and experience top-quality instruction provides, and encourage them to take more lessons wherever they ski. But at the same time, we are raising the bar on the standards ski schools must live up to.
I don't know if this is the answer you wanted to hear, Rod. But thanks for asking the question! The one thing you--and everyone else--can do to help is to complain vocally and vehemently about bad ski instruction, wherever you come across it. Insist that ski schools provide great lessons, both group and private, and refuse to accept anything less. Force them with your wallet--the only thing that really matters to the shareholders. You may not succeed in lowering the price of private lessons, but you CAN succeed in increasing the pay for instructors, because they'll have to pay more to keep top instructors on board. Improve the pay, and you'll improve the quality--and those lessons will be worth the price!