A couple of things need to be explained a little bit here. First the word insult in Greek means "I reproach, revile, upbraid." Another meaning is also disgrace. Second, the word persecute in Greek comes from the root dioko, pronounced dee-o'-ko, means "I pursue." So, in this sense Matthew 5:11 reads "Blessed are you when people disgrace you and hunt you down, and saying all kinds of evil against you because of me." Knowing this then it makes sense what Jesus is fully saying within both verses.
"Blessed are you when people disgrace you and hunt you down [to kilkl], saying all kinds of evil against you because of Me. Rejoice and be glad, for your reward in heaven is great, for in the same way they hunted down the prophets [to kill] who were befor you."
In the Western world, we think we know what persecution is, but the truth is we don't. Just because we're starting to see more of the spirit or hate and murder appear doesn't mean it's fully on us. We're getting there, however, and it's a matter of time that the persecution saints of Jesus isn't practiced alone in the Middle East, but the world over. But we can still take heart that our Lord and Savior exposed the darkness for what it is: temporary at best.
Jesus said this, "These things I have spoken to you, so that in Me you may have peace. In the world you have tribulation [no other way out], but take courage; I have overcome the world" (John 16:33). And it is in this sense that the His Church endures for a short time into the end before He comes back; and this is right at the door of the Great Tribulation.