Providing a source is really just the first step in having a serious, fact-based discussion. The next step, as antithesis says, is assessing the quality of that source and actually closely reading and discussing the information presented.
In that regard, not all news outlets are created equal. Some of them have simply a greater commitment to the truth than others. You'll find those clustered around the center of this chart. The farther you get to the edges, the more likely the news outlet's agenda will take over and you'll be presented with a skewed or outright false story.
Of note, this chart is just about news sources: Encyclopedias, history books, peer-reviewed scientific journals, etc. are generally even more reliable sources.
But they don't operate on the 24-hour news cycle. It takes additional time for those kinds of sources to be researched, written, edited, and published. They're not always available for making decisions in real-time.
So no, just because someone disagrees with your source, that's not the end of the discussion. If they bring their own valid information to the table, then you need to be prepared for the possibility that your own source is missing something or is just wrong.