Welcome, one and all, to one of the more difficult sections in the entire NT.
This passage is rife with controversy for some and confusion for many. In an overly simplified, macro-level, bird's eye view (I'm trying to qualify this as much as possible), let me offer a brief chapter summary. Chapter 9 reckons with the doctrine of election, specifically of individuals like you and me, and answers objections to that doctrine. Chapter 11 has the same doctrine of election in view, but its content is dealing with the people of Israel as a nation and whether or not God had rejected them. Chapter 10 sits in the middle with statements about faith and responsibility.
And that's where I want to focus. I love the doctrine of election. It's a precious truth to me that God is the One who chose me and set His love upon me freely and unconditionally before I did anything good or bad, making provision for me through Jesus Christ. It glorifies God for mercy since I did nothing to earn His attention or affection. It's pastorally helpful because I can look at the proud and challenge them and also look at the downtrodden and encourage them. All in all, it makes God look big and beautiful and me look small yet valued. That seems like the way it ought to be.
But some take the doctrine of election to an unbiblical extreme and, in ministry theory and practice, elevate it as the end-all and be-all of salvation. As a corrective: the CROSS AND EMPTY TOMB is the end-all and be-all of salvation. And yes, I did say "is" because the Scripture speaks of the two events as one.
And this is where Romans 10 comes in. No biblical doctrine of salvation is complete without a necessitated human response. In other words, God clearly works through means to accomplish His purposes. How is the message of the cross and empty tomb heard? It is heard through the preaching of God's Word. Who preaches? Not an angel from heaven (most of the time), but a human messenger who bears witness to the truth of it as well as the truth of it in his or her life. Through that witness, God brings light to darkened eyes so that the most rational and appropriate response by the hearer is to repent and put confidence in Christ's work at the cross and empty tomb. When hearers do so, God saves them.
There is human action and responsibility all over Romans 10. And I love that God plunked it down in the middle of two chapters on election. And I love when people take up the responsibility to witness and minister in Jesus' name. And I love when God, through their witness, saves people.
The question for you and me is this: to whom do we need to carry the message today?
But that's just me thinking thoughts...