"What I tell you in the dark, speak in the daylight ..." (Matthew 10:27)

"What I tell you in the dark, speak in the daylight; what is whispered in your ear, proclaim from the roofs." (Matthew 10:27)

What is Jesus telling them to 'speak' and 'proclaim'?

Jesus is telling his followers to pass on his teachings to others. This is confirmed with the words, "speak" and "proclaim." Both of these words indicate that Jesus is instructing his followers to speak the Truth to others.

This is a critical part of Jesus' mission. Jesus taught the same teachings of his teacher, John the Baptist along with the Prophets. And he expected his followers to do the same.

This is the process of how knowledge is passed from teacher to follower, as was followed over thousands of years through the prophets from Abraham to John the Baptist. This is the process the Supreme Being uses to pass His message through: Through devoted messengers who have learned to love and serve Him.

While Jesus may have instructed them in private, he expected them to follow those teachings and then spread those teachings to others.

Did God instruct them from within?

Yes, the Supreme Being instructed them on what to say from within. And Jesus wanted them to teach those instructions to everyone who is receptive. This was also expressed by Jesus earlier:
"But when they arrest you, do not worry about what to say or how to say it. At that time you will be given what to say, for it will not be you speaking, but the Spirit of your Father speaking through you." (Matthew 10:19-20)
This same statement was recorded slightly differently in Luke:
"When you are brought before synagogues, rulers and authorities, do not worry about how you will defend yourselves or what you will say, for the Holy Spirit will teach you at that time what you should say." (Luke 12:11)
This might bear the question: Why did Jesus' students need Jesus as their teacher if the message comes from the Supreme Being from within?

The role of the authentic spiritual teacher is to introduce the Supreme Being to the student and guide the student spiritually in practical ways, according to the time and circumstance. He also provides an example to follow. This is a living process. It is a process of relationships. Just as a friend will introduce us to their friend, the Supreme Being utilizes those who love Him to introduce us to Him.

You see, the Supreme Being is not a vague force or burning bush. He is a person - albeit the Supreme Person. The Supreme Being enjoys loving relationships.

And this is why we are attracted by loving relationships. Because we were created by the Supreme Being. We come from Him. And this is why Jesus, Moses and other representatives of God taught the same message:
"Love the Lord your God with all your heart and with all your soul and with all your mind and with all your strength.’ This is the most important commandment." (Matthew 22:37-38)