When we approach Scripture, how do we determine what the Holy Spirit is saying? Either each individual is ultimately his own highest authority regarding what the Holy Spirit is saying, or that authority belongs to something outside the individual. In the former case, we are left with “private judgment,” and the endless fragmentation that must accompany it, as history shows. But the only plausible authority outside the individual, for determining what the Holy Spirit is saying in the Scriptures, is the Church.
What is meant by "determine" is ambiguous. It would be clearer if he answered the analogous question "how does Mr. Cross determine what the Roman magisterium is saying?" Nevertheless, here are the two most likely possibilities that come to mind:
1. Perhaps the question posed desires an answer to how one can understand what a communication - specifically, the communication of God through Scripture - means. Denigrating private judgment in this sense would be a plain attack on the divinely ordained function of language in general or special revelation in particular.
Of course, since at least one of Mr. Cross' collaborators thinks that [some] Scripture is incapable of being understood apart from ecclesiastic intermediaries (link), I suppose the latter actually could be what he meant by private judgment. Suffice it to say that the Protestant position is that the ability to understand Scripture is not only available to an elite few who in turn dumb down the message for their subordinates (Psalm 19:7); that is Gnostic.
To respond to the argument in the above link, in order for the argument against perspicuity to work, Nehemiah 8:1-8 must ironically be perspicuous. But to be clear, Nehemiah 8:1-8 doesn't say exposition was necessary for the people, though such certainly could have been conducive to their understanding and is further useful in that it demonstrates clarification and deduction from Scripture is legitimate. Most important to the present subject is the point that Nehemiah 8:1-8 is actually just an example of obedience to the prescription set down in Deuteronomy 31:9-13, a prescription in which Moses makes no mention of a need to speak anything more than what was written (cf. Joshua 8:30-35, 2 Kings 23:1-3, Nehemiah 9:1-3, etc.).
2. Perhaps private judgment refers to the process by which an individual decides to which of two or more mutually exclusive interpretations - one of which may make the most sense to the individual but all of which he understands - he will submit as authoritative. In this case, one can only take Mr. Cross' word that if his own judgment conflicted with that of the Roman magisterium, he would submit to the latter; in other words, he cannot definitively demonstrate to others that he is not one of the men alluded to in 2 Timothy 4:3-4. Finally, as John Robbins wrote in Slavery and Christianity (pg. 34):
The decisions not to think about certain subjects, to abandon one's own analysis, and to trust the Roman Catholic hierarchy in matters of faith and morals are themselves decisions of private judgment - rash, uninformed, and foolish, to be sure, but still private judgment. Those individual decisions, moreover, are not made once for all; they are made daily. The decision not to think about the most important matters is a private judgment that is made repeatedly by every subject of the Roman Church-State. The attack on private judgment is not only self-contradictory, it is also un-Biblical. Speaking to the multitudes, Christ Jesus himself commands them to think for themselves rather than rely on their religious leaders and experts: "Yes, and why, even of yourselves, do you not judge what is right?" (Luke 12:57).