I was reading Stephen Greenblatt's Will in the World a biography of William Shakespeare which gives some minor background on the debate between Catholics and Anglicans. He mentions Edmund Campion who apparently after years of hiding was captured and tortured. After his torture he was pitted in a debate against Anglicans, he had no preparation at all and faced unfair odds but still utterly destroyed the competition on three separate occasions.
I was wondering what that debate or any between these two factions would have looked like. What were key points of contention between the two parties? What would their justifications have been for their positions been?
The central issue would have been one of authority. That is, who has authority to dictate the teaching and practices of the church in England?
Roman Catholics would have argued for the primacy of the Pope of Rome, based on Apostolic Succession from St. Peter.
Protestants of the time would have contested this, arguing that Peter's authority was not established by scripture to the degree that Roman Catholics stated and that the ancient church had a long practice of autocephaly, that is, the ability of individual regional or national churches to operate autonomously.
There were many other points of theological debate of course, primarily around the rights of clergy to marry vs. remain celibate, the translation of scripture and church liturgy into local languages instead of being in Latin, and most importantly, the theological principles of Lutheranism -- which became emblematic of most Protestant groups -- that faith alone and not "works" (good deeds done or alms paid) were needed for justification as Christians, an idea known as sola fide.
Most significantly of all, Protestants would have argued that Scripture was the final arbiter and source of authority in churches, and that Roman practices that were not scriptural should be reformed or ended altogether. This is known as sola scriptura
Those three basic ideas: that the Bishop of Rome did not hold primacy over all churches everywhere, sola fide and sola scriptura would have made up the foundation of any debate between Catholics and Protestants.
Obviously this is much more complicated, and would have varied depending on the time, but, returning to England, the central point of debate would have revolved around the authority of the Pope vs. the authority of the King of England to dictate how churches within England were to conduct their affairs.
As an example of the rhetoric used by Anglicans at the time, you may be interested in the "39 Articles of Religion," which are as close as Anglicans ever got to a unified statement of doctrine. They were published in 1571, and in them you can see the subtext of the controversy in statements like:
Article XIX: OF the Church: The visible Church of Christ is a congregation of faithful men, in which the pure Word of God is preached, and the Sacraments be duly ministered according to Christ's ordinance, in all those things that of necessity are requisite to the same.
As the Church of Jerusalem, Alexandria, and Antioch, have erred, so also the Church of Rome hath erred, not only in their living and manner of Ceremonies, but also in matters of Faith.
Note how the "Church of Rome" is made equivalent with other historic episcopates and, like them, also capable of error. Similarly, the Articles veer into the polemical with statements against Roman doctrine like this one:
Article XXII: Of Purgatory: The Romish Doctrine concerning Purgatory, Pardons, Worshipping and Adoration, as well of Images as of Relics, and also Invocation of Saints, is a fond thing, vainly invented, and grounded upon no warranty of Scripture, but rather repugnant to the Word of God.
Note the appeal to Scripture as a higher source of authority than "Romish Doctrine." This was a drum that Protestants were ready and willing to bang, which is part of the reason that the translation of the Bible in vernacular languages was opposed by the Roman hierarchy.
Again, I'm speaking only in generalities here, but these are the kinds of arguments that would have been common in these debates. Roman Catholics would appeal to Tradition and to the Primacy of St. Peter, Protestants would make a direct appeal to Scripture and to their interpretation of the polity of the "Early Church" -- and both sides would have had their own narratives about what the "Early Church" actually looked like.
For another primary source example of the kind of debate that was going on, you may want to read some of Luther's On the Bondage of the Will, which is his polemical response to Desiderius Erasmus' critiques of the Reformers. Conversely, you can look at the decrees that came from the Council of Trent to see how the Roman Catholic church tried to counter the Reformers.
Sources: Diarmaid MacCulloch's The Reformation, the 39 Articles
It might help if you specified a time period, given the prevalence of each denomination today - although the two people you mention (Shakespeare and Campion) lived a while apart, do you mean the 16th century in general?