Is this the case? Let's take a moment to examine Paul's discussion of sin in both Romans and Ephesians, starting with the epistle to the Romans.
One of the most extensive discussions of human depravity is found in Romans 1:18-32. However, right afterwards, we find the apostle Paul writing this:
Therefore you have no excuse, O man, every one of you who judges. For in passing judgment on another you condemn yourself, because you, the judge, practice the very same things. [Romans 2:1]Previously, Paul had been dealing with the pagan mindset of the general world. Now he turns it on the Jews. The Jews condemned Gentile society for all of those things mentioned in Romans 1, and yet, as Paul says, they were guilty of the same thing. The Jews believed that they were considered righteous by their possession of the Law, and yet, as Paul points out here and throughout most of his epistles, no one is justified by the Law or works of the Law. In fact, Paul contends that judgment from God will be carried out regardless of whether or not they follow the Law or have good works of which to boast.
For all who have sinned without the law will also perish without the law, and all who have sinned under the law will be judged by the law. [Romans 2:12]God's judgment, therefore, is complete. There is no excuse for those under the Law just as there is no excuse for those outside the Law.
Many inclusivists argue of a "third way", where a person is proclaimed innocent by God out of ignorance of the gospel, but this presupposes it is simply a lack of faith that sends one to hell rather than one's sins. It also forgets that nowhere in scripture does it speak of a third way where someone is not sent to hell simply because they never heard the gospel. There are those who are Christ or against Him (Matt 12:30) - there is no neutral ground, and lukewarm theology is an abomination to God (Rev 3:16).
In any case, Paul continues his criticism of Jewish believers who uphold their righteousness by their Jewish identity, pointing out that, as they are unable to follow the Law, they in fact bring disservice to God through their disobedience.
You who boast in the law dishonor God by breaking the law. For, as it is written, "The name of God is blasphemed among the Gentiles because of you." [Romans 2:23-24]This brings us to the climactic moment of the book of Romans:
What then? Are we Jews any better off? No, not at all. For we have already charged that all, both Jews and Greeks, are under sin, as it is written: "None is righteous, no, not one; no one understands; no one seeks for God. All have turned aside; together they have become worthless; no one does good, not even one." [Romans 3:9-12]Paul now brings the previous two and a half chapters to a hilt: everyone is sinful. Jew, Gentile - everyone. "All, both Jews and Greeks" is what Paul writes. This is not a matter of "local depravity." Paul is clearly arguing here that all of mankind is fallen. The beginning of chapter three is summarizing everything that was discussed throughout chapters one and two - and the summary is that we are all sinners in need of a savior.
Now let's turn to Paul's epistle to the Ephesians, specifically the beginning of chapter two.
And you were dead in the trespasses and sins in which you once walked, following the course of this world, following the prince of the power of the air, the spirit that is now at work in the sons of disobedience—among whom we all once lived in the passions of our flesh, carrying out the desires of the body and the mind, and were by nature children of wrath, like the rest of mankind. [Ephesians 2:1-3]Paul begins this epistle by directly addressing the Ephesian readers, stating: "You were dead in the trespasses and sins" (v. 1). It is true that he is specifically speaking to the Ephesian Gentile Christians here (we'll see that as this discussion progresses), but look at what the apostle says in the next verse: "following the course of this world" (v. 2). He calls unbelievers the "sons of disobedience", then says in verse 3: "among whom we all once lived." Who is the "we all"? It's the Jewish believers - Paul included. If one reads the first chapter of Ephesians, we see that Paul is expanding on the extension of salvation from the Jews to the Gentiles (especially Eph 1:11-13; compare to Eph 2:11-13). In many ways, Ephesians 2:1-3 is simply a longer version of Romans 3:9. However, the clincher in our discussion comes at the end of verse 3: "[we] were by nature children of wrath, like the rest of mankind." Paul says the "rest of mankind" are "children of wrath" (also translated "objects of wrath"), showing that everything he has discussed in these three verses is a statement of the spiritual condition of all of mankind.
In fact, Paul's point here is so obvious that I must be perfectly blunt: anyone who argues Paul is here teaching about a "local depravity" rather than what we might call "universal depravity" is either arguing from a second-hand source or purposefully ignoring parts of scripture. If you are guilty of the former, I encourage you to study God's word more; if you are guilty of the latter, I ask you humbly to repent of mishandling God's word.