The very premise of your question is fallacious. You are approaching with the preconception that men have such authority over God's Words. They don't. You have to establish that such men have that authority from God. And if they do where is it specifically they were given to judge what are and are not God's Inspired Words?
So start there.
The very premise of your question is fallacious. Y
This has already basically been addressed , and the answer is the same way OT writings had become established as being of God, which were invoked as substantiating Truth claims by which the NT church was established.
If a canon of any length had been established as being authoritative - and it is clear that one did by the time of Christ, as seen by the abundant clear references to OT writings in the New - and which, as with men of God, was essentially due to their distinctive enduring Divine qualities and attestation, then a future settled canon can be realized the same way.
"And on what authority" is a loaded question, as it presupposes a basis for any authority to settle the canon, and then the issue become whether such an authority must be one which cannot err in major universal faith and morality decisions.
To be sure, the OT magisterium was given binding and loosing power to settle disputes, dissent from which was a capital crime. (Dt. 17:8-13) And which corresponds to Mt. 18:15-20, though that also extends into the spiritual realm and therein to believers in general. And Westminster affirms the office of the magisterium.
And certainly it can be assumed that those who sat in the seat of Moses had an important role in establishing OT books, yet what is incontrovertible is that the "laity" discerned both men and writings as being of God without a perpetual infallible magisterium, and which is how he church began.
But which is contrary to what RCs teach, which is that an infallible magisterium is essential to correctly know what Scripture is and means.
Which is the alternative to SS (not only the typical RC strawman of it), and which is what should be the issue.
Recognition of these books as being Divine came from the same source that recognition of OT men and writings being of God came from. That being their unique Heavenly qualities and attestation. And as these writings testify to writings of God being recognized and established as being so, and thus in principal these writings provided for a canon of Scripture.
Now as concerns your alternative, Where was an infallible magisterium ever essential for men to correctly discern writings of God as being so, as RCs argue, and their meanings?