Opinions reflect feelings and may or may not be true. If one claims that their thoughts are simply "opinions" (not claiming to actually be truth- even though it might be) then they have no responsibility to provide a reasoned defense of the position.
Some people believe that no persons' ideas are more valid or true than another. All ideas are "opinions", including their own. They do not have a responsibility to defend their own ideas. Many times, when these people speak or write, you should always assume a silent prefix to their statements. That prefix is, "In my opinion". They automatically assume it for others, why should others not assume it for them? Many of these people ask others to defend their own ideas, because the others claim that their ideas do reflect truth (are not merely opinions). This is a reasonable expectation, but not from a person who claims that we only deal with opinion. Such a person believes that their opinion needs no reasoned defense (which would raise the "opinion" to the level of a "truth claim"). Keep in mind that they also believe that all opinions are equal. Thus, to be consistent, they must never require or even ask another person to defend their opinion.
If a person asks another to defend the truth of an opinion (while holding that their own opinion needs no defense), that person acknowledges that the other opinion is not the same kind of assertion. An assertion that is simply an opinion cannot (as opposed to "may not") be compared to an assertion that is a truth-claim.
In my post linked to above, I mentioned the danger of focusing on negative arguments. Opinion and the reliance on negative arguments are closely related. If one only provides negative arguments, at minimum they believe a truth-claim, but are unable (or unwilling: not sure which is worse) to defend it; at maximum they hold an opinion that they believe warrants no defense.
Many people are only willing to discuss other people's truth claims (with the sole intention of dismantling them) while refusing to allow those in the conversation to know their beliefs or reasons for their beliefs. This unwillingness to raise an opinion to the level of a truth-claim is evidence of a deliberate disconnection between their mind and their emotions (see my Psychology Class Series). If someone does not provide a defense for an opinion or even allow an opinion to be known, they have allowed an emotional attachment that is called "a priori" or "prior to the evidence" to enter their worldview. (Slight tangent- I will acknowledge that some people say that they have not made a decision with regards to certain beliefs, and that is why they do not want to commit, defend, or make known which direction they are leaning. That does not seem like a good reason to me for this reason: if you wish to focus on challenging another person's belief rather than figuring out what you believe, you are still emotionally committing to a belief: that taking down their belief is more important that building up your own. Most likely this is caused by the further belief that if your opinion is found to not accurately reflect reality, you may need to accept a truth-claim that is in direct conflict with another deeply held belief.)
Two things come to mind for the Christian here (me included). First, obviously don't just offer opinions. Provide what you believe and reasons for those beliefs. If they are challenged, honestly investigate and adjust if warranted. Second, if you are in discussion with someone who is only offering opinions or only trying to knock down your truth-claims, now is the time to exercise discernment. If these behaviors are present, that is a strong indication that the heart has not been prepared by the Holy Spirit for acceptance of the Truth. No amount of evidence will sway the person (see my post "Can You Argue Someone Into The Kingdom?"). You should change your expectations with regards to the conversation with the person, then take one of two courses. The first would be to just stop the conversation to avoid frustration and "test the water" every now and then to see if things have changed. Second would be to continue the conversation for the sake of practice, for interaction with someone who is sincerely posing the questions/challenges.
It is quite important that we understand the distinction between opinion and truth-claims. Being able to identify the equivocation in our behavior will help us to be more academically honest with ourselves in our formation of our worldview and with those around us in our conversations. Being able to identify the equivocation in others will help us to limit our frustration and even grow in our defense of our worldview.