"I'm entitled to my opinion" is something people say when they've started to get offended. The correct response to this may be "sure, sorry," or "yes, you are. Now, as I was saying..." depending on what you're talking about. It should be the former if you find yourself in some stupid argument about taste, like "chocolate ice cream is definitely better than vanilla." If it's about something with objective elements, then fair game, as long as you're concentrating on those. But it's not so easy, right?
Since my readers are mostly liberal, I'll start of with an example in which a liberal shut down a discussion because he viewed it as offensive. There was an article about "new germ theory." The author argued that many diseases we believe are genetic cannot possibly be so because anything that reduces the likelihood that you will reproduce even by 1% will be eliminated from the gene pool within x (I don't remember the number) generations. He thus argued that diseases like MS, heart disease, and many cancers were caused by viruses. (By the way, he did present actual evidence of this, too, so there's something to the line of reasoning). Whether he is right is beside the point, for the crucial part came toward the end, when he mentioned that homosexuality ought to reduce the likelihood of reproduction and therefore ought not to exist. The fact that he does, he reasoned, suggested that it came from somewhere else, perhaps some sort of infection.
The person I mentioned commented in class that he found the article interesting up until that point, but then he stopped reading. He has many gay friends, he said, and this sort of bigoted talk was simply unacceptable. Stepping through the looking glass into bizarroworld, I, the gay man, spoke up to defend the author. Why? His idea was uncomfortable, to be sure. Gays have long fought not to be seen as "sick," and here was a guy trying to throw this back at them again. But the fact that I do not like where this is going does not mean that any such sort of line of investigation should be halted. There was no suggestion that the author was using this to, say, deprive gays of their rights or lower their status. He was a rebel in the world of disease research and was thus also asking a politically incorrect, but valid, question, "Why is there homosexuality?" while attempting to rule out certain reasons and put forth a more likely explanation, as he saw it. He is not (necessarily) a homophobe for asking any more than someone asking why blacks in America earn less than other racial groups is (necessarily) racist. It's the context (how he says it and why), yes, but also the substance (what he actually says).
Moving on, an opinion like "gun control is bad" is not immune to questioning because it is not a matter of taste. Liking guns may lie at the heart of the opinion, just as a dislike of them may underlie arguments for gun control. If it were about liking guns, there would be nothing more to talk about (like chocolate vs vanilla ice cream). But there IS more to talk about, because being for or against gun control involves loads of other aspects and opinions are built on "facts," many of which may not be true. Getting to these is what a discussion is about. One person may mean "banning guns" when they think gun control, the other may not. Some may be encouraged by statistics that show crime rates falling when bans on carrying handguns are removed (this does happen), while others are more concerned about the increased likelihood of accidents in the home. Still others may conclude that, since America will never be mostly gun free like Britain due to its constitution, it is unclear how most gun control measures, which would in effect restrict gun ownership by law abiding citizens rather than criminals, would really help. NONE of these issues is an opinion. If the two people in a discussion are on the same page with all of these but still have opposing viewpoints, THAT is an opinion, and that may be the end of the story. You may have to agree to disagree because you place value on different things (a matter of preference). In most cases, all the facts are not apparent to all people (or even most of them). Discussion is necessary to bring these to the forefront.
It is possible to question someone's views while still "respecting" them. If I come to the conclusion that fewer guns would reduce deaths and that would be good, while the other person concludes that crooks have guns and she wants to be able to protect herself, while we both know the same information, then it has become a matter of taste and no further discussion is necessary (in an idealized democracy of well-informed citizens, this is where voting would settle the matter).
We do not have to "respect" a person's opinion if it involves disliking black people, or gays, or thinking women should "know their place." You are entitled to that opinion in the sense that no one can stop you from having it and no legal action can be taken against you for having or expressing it unless it manifests itself physically (liking firing a black employee, beating your wife, or inciting violence), but that doesn't mean you have a right for everyone to "respect" your opinion by not questioning it. In the end, all opinions, ideas, and beliefs are up for debate. Conducting that debate in a respectful manner is the most useful way, in my opinion (being confrontational is unlikely to get the other person to listen and just isn't, well, nice), but debating itself is never disrespectful unless it is truly a matter of taste. Obviously, though, chocolate and vanilla ice cream together is best.