Does the Bible tell us who should be interpreting its message?

Here are two verses that can form an answer to your question (at least they do it for me):

Jeremiah 17:5: "Cursed is the one who trusts in man, who draws strength from mere flesh and whose heart turns away from the Lord."

and:

John 16:13: "But when he, the Spirit of truth, comes, he will guide you into all the truth. He will not speak on his own; he will speak only what he hears, and he will tell you what is yet to come"

Regardless of who preaches the Gospel or who translates the Scripture, a child of God must seek spiritual nourishment from his heavenly Father and He will not give a stone to a child who's asking Him from bread.

2 Timothy 3:16 says

All Scripture is God-breathed and is useful for teaching, rebuking, correcting and training in righteousness

In 1 Timothy 4:12, Paul encourages Timothy saying,

Let no one look down on you because you are young, but set an example for the believers in your speech, conduct, love, faithfulness, and purity

You see, Timothy was quite young and most scholars think he began working with Paul at an age as young as 16, though this letter to him was when he was much older, but not more than 40 years of age.

This would tend to indicate that scripture is useful for teaching all, including young people. This is re-enforced by texts like Matthew 19:13-15 in which

...little children were brought to him [Jesus] for him to lay his hands on them and pray. But the disciples scolded those who brought them. But Jesus said, “Let the little children come to me and do not try to stop them, for the kingdom of heaven belongs to such as these.” And he placed his hands on them and went on his way.

Or texts Like Luke 10:39-42 which states,

Now as they went on their way, Jesus entered a certain village where a woman named Martha welcomed him as a guest. She had a sister named Mary, who sat at the Lord’s feet and listened to what he said. But Martha was distracted with all the preparations she had to make, so she came up to him and said, “Lord, don’t you care that my sister has left me to do all the work alone? Tell her to help me.” But the Lord answered her, “Martha, Martha, you are worried and troubled about many things, but one thing is needed. Mary has chosen the best part; it will not be taken away from her.”

This tends to indicate that the good news about Jesus (scripture) was intended for all men women and children.

But that's not quite your question. You did not ask who scripture was intended for but who should be interpreting it. Again, scripture gives us some important clues. For example, In Matthew 27:51 it states

At that moment the curtain of the temple was torn in two from top to bottom. The earth shook, the rocks split

You see, in antiquity, a Priest was required as a go-between between mankind and God. But God saw that this was not optimal and chose to change this system such that he was directly accessible to all through Jesus. This is why the tearing of the veil between the court where the people were and the Holy of Holies where God's throne was located. Previously, only the Priest could go before God and god spoke through priests and messengers like prophets. Furthermore, most people in antiquity were illiterate and only the wealthy, aristocrats, priests and scribes could read further re-enforcing this dependence.

Paul discusses this some in the Book of Hebrews. For example, in Chapter 4, verses 14-16 state

Therefore since we have a great high priest who has passed through the heavens, Jesus the Son of God, let us hold fast to our confession. For we do not have a high priest incapable of sympathizing with our weaknesses, but one who has been tempted in every way just as we are, yet without sin. Therefore let us confidently approach the throne of grace to receive mercy and find grace whenever we need help.

Unfortunately, do to issues with literacy and the cost of making books, Priests continued to be the sole interpretive vessel for scripture until the invention of the Printing Press and William Tyndale. People like William Tyndale, John Wycliffe and Martin Luther saw that by holding the sole power to interpret, this granted catholic priests a significant power - and power corrupts - something that can be seen over the history of the Catholic Church. This corruption became so bad, it led Martin Luther to break off from the Catholic church and start the Protestant Reformation. Ultimately, Tyndale died for his belief that Jesus should be our High Priest and we should have direct access to him. Due to this belief, Tyndale made scripture available to lay people and was burned at steak for this principle. So ultimately, all should be granted the ability to interpret.

Yet this still does not fully answer the question, because you asked, "does scripture tell us who should be interpreting scripture" and Paul's letter and the Tearing of the veil does not directly address that.

You see, despite gaining widespread literacy in modernity, there are still barriers to interpreting scripture. The Bible does in fact state who should interpret scripture in many places where it states who the texts are written to. For example, in Luke 1:1-3, it states,

Now many have undertaken to compile an account of the things that have been fulfilled among us, like the accounts passed on to us by those who were eyewitnesses and servants of the word from the beginning. So it seemed good to me as well, because I have followed all things carefully from the beginning, to write an orderly account for you, most excellent Theophilus, so that you may know for certain the things you were taught.

So clearly this text was for Theophilus to interpret. Likewise in Revelation 2:1a, it states that the letter is

To the angel of the church in Ephesus,

And in subsequent verses, it addresses sections to 6 other churches, so clearly these churches were meant to interpret this letter.

In 1 Timothy 1:2, Paul addresses his letter

To Timothy, my genuine child in the faith. Grace, mercy, and peace from God the Father and Christ Jesus our Lord!

So this letter was meant for Timothy to interpret.

In 1 Corinthians 1:2, Paul addresses his letter,

to the church of God that is in Corinth, to those who are sanctified in Christ Jesus, and called to be saints, with all those in every place who call on the name of our Lord Jesus Christ, their Lord and ours.

So clearly this letter was meant for the Church in Corinth to interpret. Yet, we return to 2 Timothy 3:16, which again says

All Scripture is God-breathed and is useful for teaching, rebuking, correcting and training in righteousness

So clearly there is both an intended primary audience, yet these texts were circulated by people like Timothy and the church at Corinth because of their usefulness in teaching, rebuking, correcting and training in righteousness. But as I mentioned earlier, there are still barriers to interpreting scripture. So what exactly are these barriers? I recently made a rather lengthy answer to a post discussing some of these barriers which I will allow you to view by clicking the link in Lieu of repeating that post here. The summary however is that so much can be lost in translation and in the delta between our worldview and frame of reference in modernity and that of Theophilus (for example).

Conclusion

This means that God would like us all to be able to interpret scripture directly, but due to practical barriers, this is not always possible. Because of the practical barriers to being directly in his Presence, God provided the High Priest for a time, but later removed that barrier so that we can seek him as directly as possible. Because of this principle, interpretation of scripture is fair game for everyone, but those that are more capable of overcoming the barriers of translation and are able to interpret scripture from the perspective of the intended primary audience of that scripture are going to be most successful at interpreting it properly. We should therefore trust their expertise over our own judgement. But as Ronald Reagan said, "Trust, but verify" as history has taught us that blindly trusting those experts can have disastrous results. This means that in order to verify what those experts, we should all be looking to better educate ourselves about scripture, translation of scripture and the historical sociology of the intended audience of scripture.