Hospital accounting is always messy. There's no simple way to determine whether a particular program is profitable or not; it comes down to how the accountants allocate overhead costs. But the fact that publicly owned and non-profit teaching hospitals aren't voluntarily expanding their residency training programs is strong evidence that they operate at a loss.

Very few other career paths have such an extensive body of knowledge, licensing requirements, and low tolerance for errors. Law is maybe a bit similar in that new associates in most firms are worse than useless, and training them sucks up a lot of time from senior associates and partners. But law firms aren't subject to price fixing, so they have more freedom to raise their rates in order to cover those costs.