That's what will happen. It has been happening for five hundred years. We will each quote Scriptures, they contradict, and if we persist, the conversation will indeed get heated.
Maybe.
But it's a shame that a productive and useful conversation may not take place because we are afraid of it getting out of control. It's a shame any time someone doesn't do something for fear that it may go wrong.
The problem is not that we don't know our faith and the Gospel. We both know it. We both can explain it to others. But what we know and what we will be explaining, is quite different. And we can't agree on it because we are both certain we are right. Writ large, this is why Christianity is divided. It cannot be reconciled either, without one capitulating to the other, because the positions are almost polar opposites.
Again, maybe I'm misunderstanding, but THE Gospel is centred on Jesus; that he, God, was born as a human, that he taught, lived, died, was raised again and ascended to heaven, 1 Corinthians 15:3-4. And that he chose to offer his perfect life as a sacrifice for the sins of the world, John 1:29; John 10:11; Matthew 26:28; Mark 10:45; Romans 5:6-8; - and many others.
It was this act which made it possible for us to be forgiven and reconciled to God. In the OT, an perfect animal was killed, and its blood shed, to make atonement for their sin. This was only temporary, however; many sins were committed so many animals were killed. Jesus came to die once for all, Hebrews 9: 28, Hebrews 10:10, 1 Peter 3:18.
The Gospel is that we were sinners and deserved to suffer the wrath of God, but that through Jesus we can be forgiven and receive eternal life and his Holy Spirit - and it is this Spirit who assures us that we are children of God.
We, in the churches, may disagree about many things, and sometimes we argue because those things are important to us.
But as far as I am aware, every denomination accepts the Gospel. Every Christian accepts and believes that it is Jesus who saves us from sin. If Jesus had not come, we would not be reconciled to God - maybe living good lives and sincerely following our own beliefs, but still sinners and apart from God. Our good deeds do not save us; our church practices and theological positions do not save us; Jesus does.
There are people who do yeoman's work trying to find the common ground,
The common ground between all denominations is, or should be, Jesus.
trying to show that, deep down, the two belief systems are the same. But they aren't.
I'm not quite sure which two belief systems we're talking about. If you mean salvation by the cross and salvation by good deeds, then you're right.
We can have a debate about the part that good deeds play in our Christian life - how does a Christian show their faith, if not by their actions - but we shouldn't try to reconcile the two. Salvation is by Jesus alone; he is the only way to the Father, John 14:6, and the only one who can save us, Acts 4:12.
So, then, the choice we each face is: What do we DO about it? Neither one of us is going to abandon our faith, what we know to be true. What does that leave, then? Fight about it endlessly and fill the air with rancor?
No; discussion is good; fighting is not.
The thing is that as far as the Bible is concerned; people either accept the Gospel or they don't. Now we are all called to proclaim and teach the Good News, we are told we may suffer for this, because people who hated Jesus will hate us too and because darkness hates the light, and we are told to be ready to answer for the hope that is in us, 1 Peter 3:15. We have the right, and duty, to explain, teach and defend the Gospel, and I can well understand people becoming passionate about the Gospel and trying desperately to persuade those who may be mocking or refusing to believe it. But ultimately, this is not our job. We proclaim the Good News; it is God who works in someone to respond and want to become a disciple of Jesus. We are responsible for how we teach and explain, but the unbeliever's response, sign of faith, willingness to listen and so on is down to God. We can pray for them, but I don't think we can make them respond to what we are saying.
I think it might be possible to have such a conversation in private, without an audience.
I'm sure we could have a conversation by messaging each other, if you like. But this debate has been started on a public forum, and there may be people reading this thread who want to know the answers.
In a nutshell, it comes down to how one weighs the words of Scripture, what trumps what.
Scripture doesn't contradict itself and there are no mistakes, so it is not about "what trumps what". If we read a verse or passage which seems to say one thing; very often there will be another verse or passage which confirms, or explains it.
Our understanding is often at fault; Scripture never is.