The Bible maintains an important distinction between the words, profession, and, confession. To profess means to declare publicly (pro- meaning "forth" as in "to speak forth"). As such, a profession (or public declaration) can be either true or false. Paul glories in the fact that Timothy had "professed a good profession before many witnesses" (1Timothy 6:12). Hebrews 4:14 exhorts the Jewish believers to whom it is written to "hold fast our profession" because a profession is what we see in others.
It can be genuine or counterfeit. This is the case of those who "profess that they know God; but in works they deny him" (Titus 1:16). Saying aloud that you love Jesus does not mean that you really love Jesus. The testimony of others is that: "Professing themselves to be wise, they became fools" (Romans 1:22).
To confess, on the other hand, means to declare with (con- meaning "together" or "with"). Basically, it means to come to the point where you admit or acknowledge that something is true. The word is used in two foundational ways in the Bible:1) to admit guilt in a sin (as in "confessing my sins" - Daniel 9:20); and, 2) to acknowledge a truth to be so (as in those who "confess that Jesus is the Son of God" - 1John 4:15).
The word by itself does not require a public statement. If it did, there would be no need for statements like, "Whosoever shall confess me before men" (Luke 12:8). Since confession involves admitting or acknowledging something that is already true, a confession is by nature true. However, the opposite is the case with a profession which can be faked or falsified.
Having this straight helps, but it does not automatically solve the problem of Romans 10:8-10. Look at the passage:
Romans 10:8 But what saith it? The word is nigh thee, even in thy mouth, and in thy heart: that is, the word of faith, which we preach; 9 That if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, thou shalt be saved. 10 For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation.
The confession of this salvation plea is the acknowledgement of the Lord Jesus (a controversy I will sidestep at this time). The word, confession, by itself does not require a public announcement. But Paul adds the phrase "with thy mouth" and this must be properly understood. This phrase does not require a public announcement in a church service or other forum, but it does point to an outward declaration, if only to the person who has told you about the salvation of God. Of course, the problem with this comes from those who because of shyness or lack of opportunity pray a silent or unheard prayer to the Lord for salvation and do not talk to someone at that time. Are they saved?
Even here, if they are truly turning to the Lord, they are saved. Their confession is primarily to the Lord, not to others. However, Romans 10:11 follows up on this very subject: "For the scripture saith, Whosoever believeth on him shall not be ashamed." That is, someone who truly believes in the Lord Jesus will not be afraid to admit that they have trusted in Him. True faith brings a kind of boldness to every believer.
Though they still may hesitate to stand before a crowd, they will certainly acknowledge their faith in a private conversation to others. Those who keep it secret even among believers are probably not saved.
So, is a public profession of faith necessary for salvation? Well, first of all, to call it a public profession is redundant. A profession is public by its very nature. There is no such thing as a private profession. Second, a profession may not even be true. Therefore, it is difficult to think of it as a requirement for salvation. Third, the confession of Romans 10:8-10 that leads to salvation is not necessarily a public one.
But this does not mean that a public confession is a bad idea. It is biblical to confess the Lord Jesus Christ to others. So, although it does not save in and of itself, it does publicly declare what happened in the heart. God's people will be blessed and sinners will be drawn. And the new believer will find the importance of not being ashamed of their faith in Christ.