What criticisms are directed towards diplomatic immunity?
Diplomatic immunity has some important uses. It goes along the line of the quote "Don't shoot the messenger." Generally, the concept is useful so that diplomats can perform their duties and maintain relationships between countries, even when those countries may be hostile towards one another.
However, this form of immunity has been the subject of constant abuse. Most of the time, this abuse centers around minor things, such as refusing to pay parking tickets. For example, in New York, the city has resorted to towing cars with diplomatic license plates that violate parking laws because parking tickets are almost always disregarded. The city reasons that even though the diplomats are immune, their cars are not!
Unfortunately, while most diplomats are honorable and reasonable people, there are some that commit significant crimes that do not have anything to do with their position, and they are able to hide behind their immunity. For example, the son of a Ghanaian diplomat was tracked down for a series of rapes by the NYPD and identified by two women. However, he walked free after 45 minutes while laughing at the women he raped because of diplomatic immunity.
Sometimes, in these cases of universally bad crimes, the diplomat's government can waive immunity and allow prosecution, which has been done in the past. For example, an deputy ambassador from Georgia (the country, not the state) had immunity waived when charged with manslaughter in America, for which he served 3 years before being paroled.
Other times, the foreign government may refuse to waive immunity, and they may try the person in their own justice system. A Russian diplomat committed manslaughter in Canada and was tried and convicted in Russia, serving his sentence in a penal colony. This latter option is more often used, too, because diplomats are often privy to state secrets, which may be extracted during a jail sentence in a foreign country.
Finally, the government that receives the diplomats has two options for removing a diplomat without violating diplomatic immunity. The first option is to declare a person "persona non grata." In other words, the diplomat would no longer be welcome in the country. Another is to expel the the entire diplomatic staff of a country, like what has been done to Syrian diplomats by many countries in the past week after another massacre by their government.
In the end, the policy is useful to prevent certain people from being unduely harassed. If two countries are negotiating a significant policy, and a diplomat is detained, some significant consequences could certainly result! However, considering that there are 37,000 people with such immunity in New York and Washington, maybe it should be more restricted!