Christian shouldn't begin to doubt their beliefs simply because of the unwavering attitude of a speaker they listen to in a debate who denies that the Christian voice is suppressed in the public by anti-Christian influences.
Some people argue a point for pay even when they know it might be harmful to others or for other self-centered interests, and these reasons are among others that don't relate to the public good.
When a Christian listens to an argument in person or on the news between someone who supports greater Christian freedom of expression and someone who has alternative views, it is important to keep in mind that people don't always argue a point based on what they personally feel is right and they won't necessarily change their position if they suddenly realize that they may be very wrong. Some people argue a point for pay even when they know it might be harmful to others or for other self-centered interests, and these reasons are among others that don't relate to the public good. It is worth keeping this in mind because some Christians may be swayed to doubt their beliefs mainly because of the resolution they see in people articulating opposing points.
Also, in a formal debate, the main objective generally isn't for each or both participants to come to a deeper understanding of an issue. Instead, it is to find the best way to support one's side, even if a debater doesn't necessarily support their side with ethical aims in mind. This objective should be kept in mind when analyzing a debate.
Truth may certainly still come out of a formal debate or argument. It is also still a good idea to listen to two opposing sides, but it needs to be kept in mind that someone may be defending a position for unethical reasons, and this is especially problematic when a speaker is able to justify a position well that ultimately isn't conducive toward greater freedom and equality for Christians and other people.