It's perfectly reasonable for the FTC to be involved here, and what constitutes unjust enrichment and anti-competitive practice when covered works are used as components in a training set is within their purview.
The USPTO will be involved in issuance questions, and the legal system will be involved in determining single cases.
On this issue, the following is my opinion (not a lawyer, not legal adv.).
A training set is a collection of representative input, which may or may not be covered works. The training set is a simple aggregation, it does not alter the works contained in any significant way, so it is not transformative. Therefore, the training set is as an omnibus of other works.
The weights of an system are a product of the input training set (if DL is used), and the parameters of the training environment. 1 is an omnibus of potentially covered works, the other is likely non-covered (but maybe patent-eligible in some cases). It is unclear if this is transformative, but it is certainly not covered by any of the existing exemptions (such as covers of songs).
Therefore, it is possible that the product of the above is a non-transformative derivative work based on one or more covered works, and therefore would have required a license or grant from the author(s) of those covered works. Note that authors often retain their ownership of UGC, and grant a service provider the ability to make that available to others in limited capacity for specific purposes. Some service providers (Like stack overflow) require that contributors license contributions under a specific license, and yet others transfer ownership as part of the terms of service.
If the training set was built from only entirely public domain works, then there wouldn't be a question, but the extension of copyright term over the years has robbed the public domain of anything even remotely useful for these kinds of cases.
On a side note, StackExchange contributions are CC BY-SA, which means that if the training input included anything from stackoverflow, the weights could be be affected by the Share-Alike requirement, which could lead to some interesting results if someone ever tried to request them. (However the attribution would probably be pretty easy, because you could pretty much say "all stackoverflow contributors" and be correct)