How are you defining "private interpretation?" Because honestly, certain uses of that particular verse could shut down much of Christianity and evangelism. Entire denominations would be forced to close their doors, because they don't subscribe to the "official" or "public" interpretation.
I've always taken that verse to mean that no single human has control over what the Scriptures mean.
As Yesua pointed out the original intent of the Scripture must be found first. What did the writer mean to convey? How did his readers understand what he was writing? Without that original intent the Scripture becomes meaningless.
This is where the allegorical method of interpretation fails. They often skip this step.
Having said that, once that original purpose is found, then one may apply the scripture to our present day situations. As it was said: the scriptures are timeless. We have a living Word.
"There is one interpretation but many applications."
When the sciptures say "the Bible is of no private interpretation," the meaning is that one man or group cannot claim an interpretation and force it upon others such as the RCC does today. They are the ones with the private interpretation (though they will deny it). Their interpretation is in the Catechism, and all must abide by it.
Baptsits have soul liberty--the right to believe what the Holy Spirit has led them to be true.
They also believe in the priesthood of the believer--that every person is a priest before God and can thus come straight before the throne of God, individually, without any other mediator.
The two are somewhat connected. As a priest before God he allows the Holy Spirit to lead him into the truths which the Lord teaches him.
But then again, he is constrained by the church that he joins. Every Baptist church ought to have a statement of faith. He must agree to that statement of faith. There is no soul liberty to be exercised there. That statement of faith is what brings unity to the local church. If you disagree with it you need to find a church that you agree with.