What on earth could it possibly mean for you to have (or not have) "a right to your opinion"?
One possibility that occurs to me is that the expression "I have a right to my opinion!" has to do with whether people will give you the last word — it's a claim to power over other people in conversation. Asserting "I have a right to my opinion" is a way of saying, "Shut up! I'm not talking about this with you any more!" Thus, to say "you have no right to your opinion" is a way of saying, "No, I won't shut up; I will go on trying to convince you that you are wrong."
Another possibility is that "I have a right to my opinion!" is a statement that one intends to continue to confidently assert a view which has been undermined by evidence or argument, without acknowledging or responding to the criticism. Thus, to say "you have no right to your opinion" is to say "you are being epistemically rude; stop it."
A third possibility is that "I have a right to my opinion!" is an assertion that some topics are too socially volatile to be exposed to much criticism. This seems to be what people mean when they bring up "the right to your opinion" in matters of religious doctrine. Thus, to say "you have no right to your opinion" is to say "I'm not going to stop publicly debunking your religion just because you don't like me doing it."
Fourth, "I have a right to my opinion!" could be a demand to not be treated worse socially by others on account of one's opinion, even if others may fear that the opinion may lead you to treat them worse. This would seem to be a demand for unilateral disarmament: "I will go on being bigoted against Blues, and I demand that Blues not treat me badly, even if they fear that I will treat them badly." Thus, to say "you have no right to your opinion" is to say "Yes, I am going to treat your opinion as evidence about your character and your future actions, and treat you accordingly."
Lastly, "I have a right to my opinion!" could be an effort to tar one's (nonviolent) critics by associating them with some sort of (violent) censors — an Inquisition, a secret police — and to rally defenders of freedom to attack those critics. Thus, to say "you have no right to your opinion" is to say "I do not pose the kind of threat that you are claiming. You have no business invoking the defense of freedom on your opinion's behalf, since freedom is not threatened. This is not a matter of 'rights'; it is a matter of conversation, argument, and evidence. Stop trying to escalate it into a matter of 'rights'."
It's also possible that "I have a right to my opinion" can mean "I have a right to enough time to assimilate new information without being told I have to think differently because someone else is sure they're right."
It might be interesting, the next time you come across someone who says "I have a right to my opinion", to ask them what they mean.
Now, it is said we all here pride ourselves on our intelligence, rationality, and moral sense. It is also said, however, that we are a fiercely independent bunch, and that we can let this pride of ours get the better of us. There have also been comments that the live communities that appear at meetups provide much more positive interactions than what goes on on this site's discussions; this might merit further investigation.
My point is; we've done a lot of research on how to do proper ethical and metaethical calculations, and on how to achieve self-empowerment and deal with our own akrasia, which is awesome. We've also done some work on matters of gender equality, which is very positive as well. But I haven't seen us do anything about the basic details of human interaction, what one would call "politeness" and "basic human decency". And I think it might be useful if we started tackling these, for our own sakes, that of those who surround us, and that of easing our mission along, which is, as I understand it so far, to save the world (from existential risk (at the hands of (unfriendly and self-modifying) artificial intelligence))).
What inspired me to propose this post was a video I just saw from Hank Green of the famed and fabled vlogbrothers. I hold these two individuals in very high esteem, and I would expect many here to share my feelings about them, on account of their values and sensibilities largely overlapping with ours; namely the sense that intelligence, knowledge and curiosity are awesome, and that intellectuals ought to use their power to help improve themselves and the world around them.
Here it is; I hope you enjoy it