I think making verbal threats and/or spitting on another person would be considered "threatening behavior" under most jurisdictions and that the person behaving like this should be subject to arrest if caught in the act. Most laws consider threatening behavior to be an act that makes another person fear injury or harm. Verbal threats and physical contact, including spitting, could certainly be construed as possibly leading to further harm. Any reasonable person would fear that a person who berates them verbally or spits on them might go further in their actions. Therefore, I think it is reasonable to have these perpetrators arrested before they cause more harm.
People who both threaten and spit on those around them should be arrested, without a doubt. Making threats toward someone, whether they actually intend on going through with them or not, is still possibly putting others in harm's way. In addition to this, spitting on people is considered harassment and should be punishable by law, especially if paired with threats.
Yes, I do think that people who make threats and spit on others should be arrested. As they saying goes, your right to swing your arms ends with my face. Similarly, free speech rights end when they interfere with another -- my right to life, liberty and the pursuit of happiness.
I think that if you threaten another human being you should be punished. There are varying degrees of punishment that should be kept in mind. Mild threats should be taken less seriously than severe threats on someone's life. Spitting on someone is gross, but it's not usually detrimental to someone's health. You should be punished for these acts, but the severity of the punishment depends on what crime took place.
Once again I stumble upon a question that requires and a mixed response. First of all, people who do spit, whether it is intentional or not, should be given a steady fine. As for threats, well... That depends on what is considered a "threat" and who it is directed toward. Every situation is different.
When done in private, that is intimidation, and when done in public, that is disorderly conduct. If the police is involved, as the picture seems to imply, that may constitute obstructing a public officer in the execution of his duty or maybe even assaulting a police officer. These actions are usually illegal and warrant arrest.