I do consider context regarding such matters, normally. And online, the norms are a very robust give and take--much as one can see in St Paul's letters and in Christ's public exhortations of Disciples and Pharisees in front of others.
And, I think the topic of an exhortation is also a factor.
If I saw someone hereon exiting a bar quite falling down drunk, I'd mention it privately.
Paul had no choice in his letters. He was physically far removed from the various churches, so putting everything in writing was his only option.
Jesus’ condemnation of the Pharisees was done principally for the benefit of the common people. The Pharisees, by and large, rejected Jesus teachings, so there was little if anything He could say to benefit them. However, His listeners could benefit enormously from His public castigation of the Pharisees. Considering that the Pharisees had the power—which they eventually used—to have Jesus killed, it must have been a sight to see.