I assume it might be a problem if you have a Patreon or something where people support you for creating the content since you can technically be considered making money off of it even though it's public domain, but what if you don't monetise it in any way and don't even reveal yourself as the creator? Would people even know who to sue?
Copyrights, in general, relate to the right of copying or reproduction.
Another’s copyright may be substantially violated and causing harm even if one doesn’t monetize on it: A free access to the work of art may mean that anyone who would otherwise pay for a copy or any sort of license will not pay. This is the reason why in most countries torrenting is illegal.
The last question is rather difficult to answer in a meaningful way: If one doesn’t get caught ever then one doesn’t get caught; if one does, then he does. I haven’t found the channel through which such unauthorized copies would enter “the public domain” and therefore it is hard to even make a guess on the probabilities of such infringement to be discovered and one getting sought damages on.
There are exemptions under “fair use” rights mostly in common law jurisdictions, and there are examples of non-enforcement or decriminalization in certain other jurisdictions under certain conditions. (See also: Is it illegal to infringe copyright if your boss or your client ordered you to do it?)
While the person asking the question may not have asked it in this manner, it seems like it's really a question about the concepts involved.
If a portion of the included work is small "enough" (and how small is small may vary), it might be fair use. But it may still be considered derivative work.
The distinction would be in how your work uses the original work.
For example, if you write a review of a book and include a few passages, it's fair use. On the other hand, if you write an action-hero novel and use a name of a well-known action hero and make allusions to their iconic adventures, it may be a derivative work even if you do not use any passages from the original work which created this fictional hero.
As the other answer pointed out, copyright is about the right to create and distribute copies. Even if you release the work anonymously, whoever distributes this unlicensed derivative work would be in violation of the copyright laws.
It wouldn't really matter that the distributor claims to not derive a direct benefit from it (such as sales). Distribution requires funds. If someone knowingly assists someone else in distributing unlicensed derivative work, they would have to comply with take-down notices by the holders of the IP of the original work. If they don't comply, they would be knowingly distributing unlicensed derivative work.
Copyright is actually a bundle of rights. One of the rights is the right to make copies of the work, or permit others to do so. Another is the right to distribute copies, for a charge or for free. Yet another is the right to create derivative works, new works based on the protected original, or, again, to permit others to do so. There are several other rights as well that are not relevant to this question.
A work can be a derivative work of more than one source, and would need permission from each source, assuming they were all still in copyright.
It is an infringement of copyright to make copies or to create or distribute a derivative work without permission, even if the infringer does not charge anything and makes no profit. The copyright owner has the right to charge, and the right to refuse permission no matter what fee is offered. Many people have the idea that it is fine to distribute a copyrighted work, or a derivative work, if only one does not charge, or does not make a profit. This is simply wrong.
The copyright holder also gets to decide whether to sue or not, and may not choose to sue if suing would cost more than the holder expects to get back in damages. But that is the holder's choice.
Whether something that includes a small amount of copyrighted content is a derivative work, or an infringing copy, depends on the detailed facts of the specific case. In the US, the concept of fair use applies to permit use of a limited amount of copyrighted content without permission. There is no clear bright line for what is or is not fair use. See this answer for a recent discussion of the concept and its limits. In other countries there are also exceptions to copyright for various purposes. These differ by country, and most are more limited than fair use.
If you create something that contains copyrighted content, then your creation is most likely a derived work. In US copyright law, you are not allowed to distribute it, free or for money, or even to create that work. However, "fair use" means there are exceptions where you will not be punished. It's still not legal, but goes unpunished.
What you can NOT do is put this work in the public domain. Only all copyright holders together can do that. And in the USA I believe it is quite difficult. There's no fair use exception, because you cannot actually do it. You can claim that you put something into the public domain, but it isn't in the public domain.
So what you want to do can become a legal nightmare for everyone involved. People would take this work, modify it, distribute it, because they rely on your claim that it is in the public domain. But that claim is wrong, so they can unknowingly commit copyright infringement, which could be expensive. On the other hand, you are seriously damaging the copyright holder.
Don't do it.