Jerry: There’s no evidence to support the fact the X works.
Tom: Well I believe that X works.
Jerry: X has been tested in scientific trials and was not found to work.
Tom: I’m entitled to my opinion.
Anyone who is not impressed by Tom’s argument has a good right not to be; it is entirely vacuous. Nevertheless, this argument is put forward frequently in debates as if it has some merit.
What do people mean when they claim to have an entitlement to an opinion? Well, the term 'entitled' is equivocal so we need to look at what is meant by the different meanings.
In the UK we have a right to free speech and free thought. This means that we can hold any opinion that we choose; however, this legal entitlement does not distinguish between valid opinion and invalid opinion. It simply means that we are entitled to an opinion no matter how right or wrong it may be.
- Epistemic - (Of, relating to, or involving knowledge)
In an epistemic sense, entitlement is an earned right. It's where a person has a right to an opinion because it is based on evidence or knowledge for example. In other words, there are good reasons for holding such an opinion. Opinions need to be justified and this distinguishes between valid and invalid opinion.
As can be seen, the two meanings of 'entitled' are quite different from each other. In fact, they are the opposite of each other.  states that we have the right to believe anything, with no regard as to whether it's true;  states that we are only entitled to opinions that we can justify, which means having good reasons for holding them.
Its use as an argumentative tactic
If we look at Tom's argument, he's using it in sense  - he does have a right to an opinion in this sense; however, he's implying that his right to an opinion somehow justifies his claim - as in sense .
Claiming a right to an opinion in sense  adds absolutely nothing to the argument. It is a complete irrelevance that does nothing to resolve the disagreement. Tom may as well have pointed out that he disagrees with Jerry because he doesn't like Monday mornings! Changing the subject of the argument to whether one is entitled to an opinion merely introduces an irrelevance: the 'red herring' fallacy.
If Tom was claiming that he's entitled to an opinion because he has good reasons for holding it, as in sense , then his claim carries more weight. Of course Jerry may also feel justified in claiming an entitlement to his opinion too. If their views differ, then one (possibly both) of them is wrong. If two people claim to be entitled to their opinion , how can the argument be resolved? By examining both of their arguments and finding out which has the best case to support their opinion. In other words, by resolving the original argument.
You're entitled to your opinion
Tom: Homeopathy works.
Jerry: No, quality clinical trials have shown that it's no better than placebo.
Tom: I know many people for whom it's worked; and it's worked for me too.
Jerry: There are no ingredients in homeopathic remedies so they can't work!
Tom: Well, you're entitled to your opinion.
Again, Tom is using the fact that a person is entitled to an opinion as an argumentative tactic but this time in completely the opposite manner - he's invoking its sense  usage to dismiss Jerry's argument.
This usage is a rather condescending way of someone telling another person that they don't agree with them as it includes the assumption that the person saying it is right (perhaps without having even stated their side of the argument) and they know the other person is wrong. It's nothing more than a statement of presumption and is the Bare Faced Assertion fallacy (I'm right, you're wrong - because I say so) therefore it is also a complete irrelevance in a debate.
The irony in using it this way is that it contradicts the sense  usage that Tom used to support his original argument that X works.
If it's used in this manner to dismiss a person's descriptive or factual claim then it is an instance of the subjectivist fallacy: treating a fact as if it were a subjective opinion.
In this case, the sense  usage is treated as if it's a sense  instance.
"Entitled to an opinion" is used in debates to add weight to a person's argument or to dismiss as opponent's argument.
Whether it's used to support one's position (inferring sense  from sense ), reject an opponent's argument (invoking sense ), or to add authority to a person's position (invoking sense ), it is equally useless as a debating tactic.
Debates can only be resolved by presenting sound arguments with supporting evidence. Stating one's rights and entitlements adds nothing to the debating process.
Having a right to an opinion does not make that opinion right.