A person that is modest is also reserved in speech. This means that we should not just say what pops in our minds. A lot of things may come into our mind but we do not have to say it. This is why the Bible stated that we should study to show ourselves quiet.
Also just because something is true does not mean that we have to say it. A lot of Christians and others hide behind the scripture:
And ye shall know the truth, and the truth shall make you free.
They think this gives them license to say mean but truthful things. For example, just because someone is overweight doesn't mean that we should call them fat or ugly. First it is not our place to pass judgement nor is it to hurt someone's feelings. We should speak things that build people up and not tear people down.
1 Thessalonians 5:11 [Full Chapter]
Therefore comfort each other and edify one another, just as you also are doing.
I know the Bible also mentions how we should joke or make fun of others.
I have witnessed a lot of Christians think it is okay to joke at the expense of others. Christians have often have offended people but do not see it as being wrong because it is a small sin right? No any sin is sin and should not be done. They also believe that if everyone (except that person the are making fun of) is laughing that it is okay. But just because a lot of people are being rude does not give us the opportunity to do the same.
If anyone among you thinks he is religious, and does not bridle his tongue but deceives his own heart, this one’s religion is useless.
Our communication is a reflection of us. What we speak shows what is in our heart.
Modesty is about being a good person that respects others.
So is your speech respectful for others.